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From Coq Require Import ssreflect ssrfun ssrbool.

Require Import Setoid.
From Coq Require Import Ensembles.
From Coq.Logic Require Import FunctionalExtensionality.
From Coq.Logic Require Import PropExtensionality ClassicalFacts.


New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]


From MatchingLogic Require Import
  Syntax
  Semantics
  DerivedOperators_Syntax
  DerivedOperators_Semantics
  monotonic
  Utils.Lattice
  Utils.stdpp_ext
  IndexManipulation
.

Import MatchingLogic.Syntax.Notations.
Import MatchingLogic.Substitution.Notations.
Import MatchingLogic.DerivedOperators_Syntax.Notations.


Section with_signature.
  Context {Ξ£ : Signature}.
  Open Scope ml_scope.

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  let Sfix := eval ρ (mu , Ο•) in F Sfix = Sfix

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  let Sfix := eval ρ (mu , Ο•) in F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
β†’ Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)
    (eval ρ (mu , Ο•)) = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
β†’ Fassoc ρ Ο•^{svar:0↦X} X (eval ρ (mu , Ο•)) =
  eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
β†’ F (eval ρ (mu , Ο•)) = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
F Sfix = Sfix


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction F
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction F
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction (Fassoc ρ Ο•^{svar:0↦X} X)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction
  (Ξ» S : propset M,
     eval (update_svar_val X S ρ) Ο•^{svar:0↦X})
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
well_formed_positive (mu , Ο•)
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in X Ο•
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in X Ο•

      
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in (fresh_svar Ο•) Ο•
 apply set_svar_fresh_is_fresh.
    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
F Sfix = Sfix


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
isFixpoint F (LeastFixpointOf F)
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: isFixpoint F (LeastFixpointOf F)
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
isFixpoint F (LeastFixpointOf F)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
MonotonicFunction F
 apply HFmono.
    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: isFixpoint F (LeastFixpointOf F)
F Sfix = Sfix


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
(let X := fresh_svar Ο• in
 LeastFixpointOf
   (Ξ» S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' Ο•^{svar:0↦X}))
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval
       (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val X S ρ) Ο•^{svar:0↦X})
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval
       (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqSfix: Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval
       (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Ffix: F (LeastFixpointOf F) = LeastFixpointOf F
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
HeqSfix: Sfix = LeastFixpointOf F
F Sfix = Sfix

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Sfix: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
HeqSfix: Sfix = LeastFixpointOf F
Ffix: F Sfix = Sfix
F Sfix = Sfix

    apply Ffix.
  Qed.


  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  let Sfix := eval ρ (mu , Ο•) in F S βŠ† S β†’ Sfix βŠ† S

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  let Sfix := eval ρ (mu , Ο•) in F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
β†’ Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) S
  βŠ† S β†’ eval ρ (mu , Ο•) βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
β†’ Fassoc ρ Ο•^{svar:0↦X} X S βŠ† S β†’ eval ρ (mu , Ο•) βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F S βŠ† S β†’ eval ρ (mu , Ο•) βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο• β†’ F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
F S βŠ† S β†’ Sfix βŠ† S


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction F
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction F
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction (Fassoc ρ Ο•^{svar:0↦X} X)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
MonotonicFunction
  (Ξ» S : propset M,
     eval (update_svar_val X S ρ) Ο•^{svar:0↦X})
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
well_formed_positive (mu , Ο•)
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in X Ο•
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in X Ο•

      
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
svar_is_fresh_in (fresh_svar Ο•) Ο•
 apply set_svar_fresh_is_fresh.
    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
F S βŠ† S β†’ Sfix βŠ† S


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
LeastFixpointOf F = Sfix
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
F S βŠ† S β†’ Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
LeastFixpointOf F = Sfix
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•}
     (fresh_svar Ο•))
LeastFixpointOf
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)) =
eval ρ (mu , Ο•)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•}
     (fresh_svar Ο•))
LeastFixpointOf
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)) =
(let X := fresh_svar Ο• in
 LeastFixpointOf
   (Ξ» S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' Ο•^{svar:0↦X}))
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•}
     (fresh_svar Ο•))
LeastFixpointOf
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)) =
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction
  (Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•}
     (fresh_svar Ο•))
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•}) =
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
 reflexivity. 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
F S βŠ† S β†’ Sfix βŠ† S


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S βŠ† S
Sfix βŠ† S


    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S βŠ† S
Hleast: leq (F S) S β†’ leq (LeastFixpointOf F) S
Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S βŠ† S
Hleast: F S βŠ† S β†’ LeastFixpointOf F βŠ† S
Sfix βŠ† S
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S βŠ† S
Hleast: LeastFixpointOf F βŠ† S
Sfix βŠ† S

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
S: propset M
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix: propset M
HeqSfix: Sfix = eval ρ (mu , Ο•)
OS:= PropsetOrderedSet M: OrderedSet (propset M)
L:= PowersetLattice M: CompleteLattice (propset M)
Hwfp: no_negative_occurrence_db_b 0 Ο• &&
well_formed_positive Ο•
HFmono: MonotonicFunction F
Hlfp: LeastFixpointOf F = Sfix
Hincl: F S βŠ† S
Hleast: Sfix βŠ† S
Sfix βŠ† S
 apply Hleast.
  Qed.

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  F Sfix βŠ† Sfix
  β†’ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
    β†’ Sfix = eval ρ (mu , Ο•)

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  F Sfix βŠ† Sfix
  β†’ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
    β†’ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
let X := fresh_svar Ο• in
let F := Fassoc ρ Ο•^{svar:0↦X} X in
F Sfix βŠ† Sfix
β†’ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
  β†’ Sfix = eval ρ (mu , Ο•)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) Sfix
βŠ† Sfix
β†’ (βˆ€ S : propset M,
     Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) S
     βŠ† S β†’ Sfix βŠ† S) β†’ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
Fassoc ρ Ο•^{svar:0↦X} X Sfix βŠ† Sfix
β†’ (βˆ€ S : propset M,
     Fassoc ρ Ο•^{svar:0↦X} X S βŠ† S β†’ Sfix βŠ† S)
  β†’ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
F Sfix βŠ† Sfix
β†’ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
  β†’ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
Sfix =
(let X := fresh_svar Ο• in
 LeastFixpointOf
   (Ξ» S : propset M,
      let ρ' := update_svar_val X S ρ in
      eval ρ' Ο•^{svar:0↦X}))
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val X S ρ) Ο•^{svar:0↦X})
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
Sfix =
LeastFixpointOf
  (Ξ» S : propset M,
     eval (update_svar_val (fresh_svar Ο•) S ρ)
       Ο•^{svar:0↦fresh_svar Ο•})
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
Sfix = LeastFixpointOf F

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
leq (F Sfix) Sfix
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
βˆ€ x : propset M, leq (F x) x β†’ leq Sfix x
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
leq (F Sfix) Sfix
 apply Hprefix. 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
Hprefix: F Sfix βŠ† Sfix
Hleast: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
HeqF: F =
(Ξ» S : propset M,
   eval (update_svar_val (fresh_svar Ο•) S ρ)
     Ο•^{svar:0↦fresh_svar Ο•})
βˆ€ x : propset M, leq (F x) x β†’ leq Sfix x
 apply Hleast.
  Qed.

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  F Sfix βŠ† Sfix
  ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
  ↔ Sfix = eval ρ (mu , Ο•)

  
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
well_formed_positive (mu , Ο•)
β†’ let X := fresh_svar Ο• in
  let F := Fassoc ρ Ο•^{svar:0↦X} X in
  F Sfix βŠ† Sfix
  ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
  ↔ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
let X := fresh_svar Ο• in
let F := Fassoc ρ Ο•^{svar:0↦X} X in
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
↔ Sfix = eval ρ (mu , Ο•)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) Sfix
βŠ† Sfix
∧ (βˆ€ S : propset M,
     Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) S
     βŠ† S β†’ Sfix βŠ† S) ↔ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
Fassoc ρ Ο•^{svar:0↦X} X Sfix βŠ† Sfix
∧ (βˆ€ S : propset M,
     Fassoc ρ Ο•^{svar:0↦X} X S βŠ† S β†’ Sfix βŠ† S)
↔ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
↔ Sfix = eval ρ (mu , Ο•)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
↔ Sfix = Sfix'

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
β†’ Sfix = Sfix'
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
Sfix = Sfix'
β†’ F Sfix βŠ† Sfix
  ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)

    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
β†’ Sfix = Sfix'
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H1: F Sfix βŠ† Sfix
H2: βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
Sfix = Sfix'
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
H2: βˆ€ S : propset M,
  Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
    (fresh_svar Ο•) S βŠ† S β†’ 
  Sfix βŠ† S
H1: Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) Sfix βŠ† Sfix
Sfix = eval ρ (mu , Ο•)

      auto using eval_mu_if_lfp.
    
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
Sfix = Sfix'
β†’ F Sfix βŠ† Sfix
  ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
F Sfix βŠ† Sfix ∧ (βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
F Sfix βŠ† Sfix
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S

      
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
F Sfix βŠ† Sfix
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)
  (eval ρ (mu , Ο•)) βŠ† eval ρ (mu , Ο•)

        
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)
  (eval ρ (mu , Ο•)) = eval ρ (mu , Ο•)
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
H: Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) =
eval ρ (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) βŠ† 
eval ρ (mu , Ο•)

        
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
well_formed_positive (mu , Ο•)
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
H: Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) =
eval ρ (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) βŠ† 
eval ρ (mu , Ο•)
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
H: Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) =
eval ρ (mu , Ο•)
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•)
  (eval ρ (mu , Ο•)) βŠ† eval ρ (mu , Ο•)

        
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
H: Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} 
  (fresh_svar Ο•) (eval ρ (mu , Ο•)) =
eval ρ (mu , Ο•)
eval ρ (mu , Ο•) βŠ† eval ρ (mu , Ο•)
 apply reflexivity.
      
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
βˆ€ S : propset M, F S βŠ† S β†’ Sfix βŠ† S
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Sfix: propset M
Hwfp: well_formed_positive (mu , Ο•)
X: svar
HeqX: X = fresh_svar Ο•
F: propset M β†’ propset M
HeqF: F = Fassoc ρ Ο•^{svar:0↦X} X
Sfix': propset M
HeqSfix': Sfix' = eval ρ (mu , Ο•)
H: Sfix = Sfix'
S: propset M
F S βŠ† S β†’ Sfix βŠ† S
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
S: propset M
Fassoc ρ Ο•^{svar:0↦fresh_svar Ο•} (fresh_svar Ο•) S βŠ† S
β†’ eval ρ (mu , Ο•) βŠ† S
 
Ξ£: Signature
M: Model
ρ: Valuation
Ο•: Pattern
Hwfp: well_formed_positive (mu , Ο•)
S: propset M
well_formed_positive (mu , Ο•)
 apply Hwfp.
  Qed.

  (* mu X. base \/ step X *)
  (* [Nats] = mu X. 0 \/ succ X *)
  (* [Nats] = \{ x | \ex x0,x1,..x_n . x0 \in 0 /\ x(i+1) \in succ xi }*)
  (*  0, 1, 2,... x*)
  Section inductive_generation.
    Context (base step : Pattern).

    Let patt_ind_gen_body := (patt_or (nest_mu base) (patt_app (nest_mu step) (patt_bound_svar 0))).
    Let patt_ind_gen_simple_body := (patt_or base (patt_app step (patt_free_svar (fresh_svar patt_ind_gen_body)))).

    Definition patt_ind_gen := patt_mu patt_ind_gen_body.

    Hypothesis (Hwfpbase : well_formed_positive base).
    Hypothesis (Hwfpstep : well_formed_positive step).

    
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive patt_ind_gen

    
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive patt_ind_gen

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
well_formed_positive (mu , patt_ind_gen_body)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive (nest_mu base) && true,
    well_formed_positive (nest_mu step)
  & true]

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive (nest_mu base)
  & well_formed_positive (nest_mu step)]

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    well_formed_positive base
  & well_formed_positive step]

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    true
  & well_formed_positive step]

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 patt_ind_gen_body,
    true
  & true]

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
no_negative_occurrence_db_b 0 patt_ind_gen_body


      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& (fix no_negative_occurrence_db_b
       (dbi : db_index) (Ο• : Pattern) {struct Ο•} :
         bool :=
       match Ο• with
       | ϕ₁ $ Ο•β‚‚ =>
           no_negative_occurrence_db_b dbi ϕ₁ &&
           no_negative_occurrence_db_b dbi Ο•β‚‚
       | ϕ₁ ---> Ο•β‚‚ =>
           no_positive_occurrence_db_b dbi ϕ₁ &&
           no_negative_occurrence_db_b dbi Ο•β‚‚
       | ex , Ο•' => no_negative_occurrence_db_b dbi Ο•'
       | mu , Ο•' =>
           no_negative_occurrence_db_b (S dbi) Ο•'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (Ο• : Pattern) {struct Ο•} :
         bool :=
       match Ο• with
       | patt_bound_svar n =>
           if decide (n = dbi) then false else true
       | ϕ₁ $ Ο•β‚‚ =>
           no_positive_occurrence_db_b dbi ϕ₁ &&
           no_positive_occurrence_db_b dbi Ο•β‚‚
       | ϕ₁ ---> Ο•β‚‚ =>
           no_negative_occurrence_db_b dbi ϕ₁ &&
           no_positive_occurrence_db_b dbi Ο•β‚‚
       | ex , Ο•' => no_positive_occurrence_db_b dbi Ο•'
       | mu , Ο•' =>
           no_positive_occurrence_db_b (S dbi) Ο•'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) 0 (nest_mu base) &&
    true, no_negative_occurrence_db_b 0 (nest_mu step)
  & true]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& no_negative_occurrence_db_b 0 (nest_mu base) &&
    true, no_negative_occurrence_db_b 0 (nest_mu step)
  & true]


      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
[&& (if decide (0 < 0)
     then no_negative_occurrence_db_b 0 base
     else
      if decide (0 < 0 + 1)
      then true
      else no_negative_occurrence_db_b (0 - 1) base) &&
    true,
    if decide (0 < 0)
    then no_negative_occurrence_db_b 0 step
    else
     if decide (0 < 0 + 1)
     then true
     else no_negative_occurrence_db_b (0 - 1) step
  & true]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
true

      auto.
    Qed.

    
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
svar_is_fresh_in X patt_ind_gen_body
β†’ eval ρ patt_ind_gen_body^{svar:0↦X} =
  eval ρ (base or step $ patt_free_svar X)

    
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
svar_is_fresh_in X patt_ind_gen_body
β†’ eval ρ patt_ind_gen_body^{svar:0↦X} =
  eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: svar_is_fresh_in X patt_ind_gen_body
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X βˆ‰ free_svars patt_ind_gen_body
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X
βˆ‰ free_svars (nest_mu base) βˆͺ βˆ…
  βˆͺ (free_svars (nest_mu step) βˆͺ βˆ…)
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: X βˆ‰ free_svars base βˆͺ free_svars step
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr: (X βˆ‰ free_svars base) ∧ X βˆ‰ free_svars step
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ patt_ind_gen_body^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)


      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ
  (nest_mu base or nest_mu step $ BoundVarSugar.B0)^{svar:0↦X} =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ
  (nest_mu base or nest_mu step $ BoundVarSugar.B0)^[svar:0↦
  patt_free_svar X] =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ
  ((nest_mu base)^[svar:0↦patt_free_svar X]
   or (nest_mu step)^[svar:0↦patt_free_svar X] $
      BoundVarSugar.B0^[svar:0↦patt_free_svar X]) =
eval ρ (base or step $ patt_free_svar X)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ
  ((nest_mu base)^[svar:0↦patt_free_svar X]
   or (nest_mu step)^[svar:0↦patt_free_svar X] $
      patt_free_svar X) =
eval ρ (base or step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ (nest_mu base)^[svar:0↦patt_free_svar X]
βˆͺ eval ρ
    ((nest_mu step)^[svar:0↦patt_free_svar X] $
     patt_free_svar X) =
eval ρ base βˆͺ eval ρ (step $ patt_free_svar X)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ
  (nest_mu_aux 0 1 base)^[svar:0↦patt_free_svar X]
βˆͺ eval ρ
    ((nest_mu_aux 0 1 step)^[svar:0↦patt_free_svar X] $
     patt_free_svar X) =
eval ρ base βˆͺ eval ρ (step $ patt_free_svar X)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
X: svar
Hfr1: X βˆ‰ free_svars base
Hfr2: X βˆ‰ free_svars step
eval ρ base βˆͺ eval ρ (step $ patt_free_svar X) =
eval ρ base βˆͺ eval ρ (step $ patt_free_svar X)

      reflexivity.
    Qed.
    
    
    Section with_eval.
      Context (M : @Model Ξ£).
      Context (ρ : @Valuation Σ M).


      Let F := let X := fresh_svar patt_ind_gen_body in
               @Fassoc Ξ£ M ρ (patt_ind_gen_body^{svar: 0 ↦ X}) X.
      (*
      Lemma svar_open_patt_ind_gen_body_assoc S:
        let X := fresh_svar patt_ind_gen_body in
        eval ρₑ (update_svar_val X S ρₛ) (svar_open 0 X patt_ind_gen_body)
        = F S.
      Proof. reflexivity.
             (*
        cbv zeta.
        rewrite svar_open_patt_ind_gen_body_simpl.
        { apply set_svar_fresh_is_fresh. }
        subst F. unfold Fassoc.
        rewrite svar_open_patt_ind_gen_body_simpl.
        { apply set_svar_fresh_is_fresh.  }
        reflexivity.*)
      Qed.
       *)

      (* I can imagine this lemma to be proven automatically. *)
      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
F =
(Ξ» A : propset M,
   eval ρ base βˆͺ app_ext (eval ρ step) A)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
F =
(Ξ» A : propset M,
   eval ρ base βˆͺ app_ext (eval ρ step) A)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Fassoc ρ
  patt_ind_gen_body^{svar:0↦fresh_svar
                              patt_ind_gen_body}
  (fresh_svar patt_ind_gen_body) =
(Ξ» A : propset M,
   eval ρ base βˆͺ app_ext (eval ρ step) A)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
(Ξ» S : propset M,
   eval
     (update_svar_val (fresh_svar patt_ind_gen_body) S
        ρ)
     patt_ind_gen_body^{svar:0↦fresh_svar
                                 patt_ind_gen_body}) =
(Ξ» A : propset M,
   eval ρ base βˆͺ app_ext (eval ρ step) A)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
βˆ€ x : propset M,
  eval
    (update_svar_val (fresh_svar patt_ind_gen_body) x
       ρ)
    patt_ind_gen_body^{svar:0↦fresh_svar
                                patt_ind_gen_body} =
  eval ρ base βˆͺ app_ext (eval ρ step) x

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  patt_ind_gen_body^{svar:0↦fresh_svar
                              patt_ind_gen_body} =
eval ρ base βˆͺ app_ext (eval ρ step) A
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  patt_ind_gen_body
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  (base
   or step $
      patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  patt_ind_gen_body
 apply set_svar_fresh_is_fresh. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  (base
   or step $
      patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base βˆͺ app_ext (eval ρ step) A


        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
βˆͺ eval
    (update_svar_val (fresh_svar patt_ind_gen_body) A
       ρ)
    (step $
     patt_free_svar (fresh_svar patt_ind_gen_body)) =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
βˆͺ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step)
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ)
       (patt_free_svar (fresh_svar patt_ind_gen_body))) =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
βˆͺ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step)
    (svar_valuation
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) (fresh_svar patt_ind_gen_body)) =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval
  (update_svar_val (fresh_svar patt_ind_gen_body) A ρ)
  base
βˆͺ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval ρ base
βˆͺ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
free_svars base βŠ† free_svars ?Ο•β‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 apply set_svar_fresh_is_fresh. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
free_svars base βŠ† free_svars patt_ind_gen_body

          solve_free_svars_inclusion 5.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval ρ base
βˆͺ app_ext
    (eval
       (update_svar_val (fresh_svar patt_ind_gen_body)
          A ρ) step) A =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval ρ base βˆͺ app_ext (eval ρ step) A =
eval ρ base βˆͺ app_ext (eval ρ step) A

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
free_svars step βŠ† free_svars ?Ο•β‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 apply set_svar_fresh_is_fresh. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
free_svars step βŠ† free_svars patt_ind_gen_body

          solve_free_svars_inclusion 5.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
A: propset M
eval ρ base βˆͺ app_ext (eval ρ step) A =
eval ρ base βˆͺ app_ext (eval ρ step) A

        reflexivity.
      Qed.

      Definition is_witnessing_sequence_old (m : Domain M) (l : list (Domain M)) :=
        (last l = Some m) /\
        (match l with
         | [] => False
         | mβ‚€::ms => (mβ‚€ ∈ @eval Ξ£ M ρ base)
                     /\  (@Forall _
                                  (Ξ» (x : (Domain M) * (Domain M)),
                                   let (old, new) := x in
                                   new ∈ 
                                  app_ext
                                    (@eval Σ M ρ step)
                                    {[ old ]}
                                 )
                                 (zip (mβ‚€::ms) ms)
                         )

         end).

      Definition is_witnessing_sequence (m : Domain M) (l : list (Domain M)) :=
        (βˆƒ lst, last l = Some lst /\ lst ∈ @eval Ξ£ M ρ base)
          /\
          hd_error l = Some m
          /\
          ((@Forall _ (uncurry (Ξ» new old,
                              new ∈
                     app_ext
                       (@eval Σ M ρ step)
                       {[old]}
                    ))
                    (zip l (tail l))
          )).

      (* If we have a witnessing sequence x₁ xβ‚‚ ... xβ‚™ xβ‚™β‚Šβ‚ ... xlast
         and xβ‚™ matches `base`, then xβ‚™ xβ‚™β‚Šβ‚ is a witnessing sequence, too.
       *)
      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m': M
is_witnessing_sequence m l
β†’ l !! n = Some m'
  β†’ m' ∈ eval ρ base
    β†’ is_witnessing_sequence m' (drop n l)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m': M
is_witnessing_sequence m l
β†’ l !! n = Some m'
  β†’ m' ∈ eval ρ base
    β†’ is_witnessing_sequence m' (drop n l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_witnessing_sequence m' (drop n l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
βˆƒ lst : M,
  last (drop n l) = Some lst ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (drop n l) (tail (drop n l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
βˆƒ lst : M,
  last (drop n l) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last (drop n l) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last (drop n l) = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some lst = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some lst = Some lst
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
last l = Some lst
 apply Hlst. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
n < length l
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_Some (l !! n)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
is_Some (Some m')
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Some m' = Some m'
 reflexivity.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
lst ∈ eval ρ base

          apply Hbase.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (drop n l) (tail (drop n l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
head (drop n l) = Some m'
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
l !! n = Some m'
 apply Hm'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (tail (drop n l)))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (drop n l) (drop n (tail l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (drop n (zip l (tail l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
n: nat
m', lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hm': l !! n = Some m'
Hbase': m' ∈ eval ρ base
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))

          apply Hfa.
        }
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m': M
is_witnessing_sequence m l
β†’ head (tail l) = Some m'
  β†’ is_witnessing_sequence m' (tail l)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m': M
is_witnessing_sequence m l
β†’ head (tail l) = Some m'
  β†’ is_witnessing_sequence m' (tail l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m': M
Hw: is_witnessing_sequence m l
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m': M
Hw: (βˆƒ lst : M, last l = Some lst ∧ lst ∈ eval ρ base)
∧ head l = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip l (tail l))
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
is_witnessing_sequence m' (tail l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
(βˆƒ lst : M,
   last (tail l) = Some lst ∧ lst ∈ eval ρ base)
∧ head (tail l) = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (tail l) (tail (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
βˆƒ lst : M,
  last (tail l) = Some lst ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (tail l) (tail (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
βˆƒ lst : M,
  last (tail l) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst ∧ lst ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last (tail l) = Some lst

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last l = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some ?m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
last l = Some lst
 apply Hlst1.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (tail l) (tail (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (tail l) (tail (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
head (tail l) = Some m'
 apply Hhead. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (tail l) (tail (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (tail (zip l (tail l)))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw: head l = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
m', lst: M
Hlst1: last l = Some lst
Hlst2: lst ∈ eval ρ base
Hw1: head l = Some m
Hw2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
Hhead: head (tail l) = Some m'
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))

        apply Hw2.
      Qed.
      

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)


        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
βˆ€ x : M * M,
  uncurry
    (flip
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x =
  (let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
x₁, xβ‚‚: M
uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) (x₁, xβ‚‚) =
(xβ‚‚ ∈ app_ext (eval ρ step) {[x₁]})

          reflexivity.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
↔ is_witnessing_sequence_old m (reverse l)


        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
β†’ is_witnessing_sequence_old m (reverse l)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence_old m (reverse l)
β†’ is_witnessing_sequence m l

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence m l
β†’ is_witnessing_sequence_old m (reverse l)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last l = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head l = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
is_witnessing_sequence_old m (reverse l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head [] = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse [])
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head (x :: l) = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head [] = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse [])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last [] = Some m'
Hbase: m' ∈ eval ρ base
Hhd: None = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
is_witnessing_sequence_old m (reverse [])
 inversion Hhd. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: head (x :: l) = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: Some x = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
is_witnessing_sequence_old m (reverse (x :: l))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (x :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hhd: Some x = Some m
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))
H0: x = m
is_witnessing_sequence_old m (reverse (m :: l))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Hhd: Some m = Some m
is_witnessing_sequence_old m (reverse (m :: l))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
is_witnessing_sequence_old m (reverse (m :: l))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse (m :: l)) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
match reverse (m :: l) with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse (m :: l)) = Some m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
last (reverse l ++ [m]) = Some m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Some m = Some m

            reflexivity.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
match reverse (m :: l) with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end


          (* reverse (m::l) <> nil *)
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
False
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
False
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length (reverse (m :: l)) = length []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (reverse (m :: l)) = length []
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length (reverse (m :: l)) = length []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
length [] = length []
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (reverse (m :: l)) = length []
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: length (m :: l) = length []
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Heq: reverse (m :: l) = []
H: S (length l) = 0
False
 inversion H.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
m' = d
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
Hm'd: m' = d
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
H: last (m :: l) = head (reverse l ++ [m])
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse l ++ [m] = d :: l0
H: Some m' = head (reverse l ++ [m])
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = head (reverse l ++ [m])
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = head (d :: l0)
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = Some d
m' = d

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some m' = Some d
H1: m' = d
d = d
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: rev_append l [] ++ [m] = d :: l0
H: Some d = Some d
d = d

            reflexivity.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
m': M
Hlast: last (m :: l) = Some m'
Hbase: m' ∈ eval ρ base
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
d: M
l0: list M
Heq: reverse (m :: l) = d :: l0
Hm'd: m' = d
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
d ∈ eval ρ base
 apply Hbase. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hbase: d ∈ eval ρ base
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': reverse (reverse (m :: l)) = reverse (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
l0 = tail (d :: l0)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
l0 = tail (d :: l0)
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) l0)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Heq'': l0 = tail (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) (tail (d :: l0)))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l0) (tail (d :: l0)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
l0: list M
Heq: reverse (m :: l) = d :: l0
Heq': m :: l = reverse (d :: l0)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hlast: last (m :: l) = Some d
Hfa: Forall
  (uncurry
     (flip
        (Ξ» new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hfa: Forall
  (uncurry
     (flip
        (Ξ» new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
d: M
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l))
     (tail (reverse (m :: l))))
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (reverse (m :: l)) (tail (reverse (m :: l))))

          apply Hfa.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
is_witnessing_sequence_old m (reverse l)
β†’ is_witnessing_sequence m l
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
H: is_witnessing_sequence_old m (reverse l)
is_witnessing_sequence m l

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse l) = Some m
H2: match reverse l with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m l

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse []) = Some m
H2: match reverse [] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse (x :: l)) = Some m
H2: match reverse (x :: l) with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m (x :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse []) = Some m
H2: match reverse [] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: None = Some m
H2: match reverse [] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m []
 inversion Hlast. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Hlast: last (reverse (x :: l)) = Some m
H2: match reverse (x :: l) with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence m (x :: l)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Heq: reverse (x :: l) = []
Hlast: last [] = Some m
H2: False
is_witnessing_sequence m (x :: l)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
H2: y ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
Heq: reverse (x :: l) = []
Hlast: last [] = Some m
H2: False
is_witnessing_sequence m (x :: l)
 inversion H2. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
H2: y ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Heq: reverse (x :: l) = y :: ys
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
is_witnessing_sequence m (x :: l)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: reverse (x :: l) = reverse (reverse (y :: ys))
is_witnessing_sequence m (x :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
is_witnessing_sequence m (x :: l)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: reverse (x :: l) = reverse (reverse (y :: ys))
Inj eq eq reverse

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
is_witnessing_sequence m (x :: l)

          
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
βˆƒ lst : M,
  last (x :: l) = Some lst ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (x :: l) (tail (x :: l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
βˆƒ lst : M,
  last (x :: l) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (x :: l) = Some y ∧ y ∈ eval ρ base

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (x :: l) = Some y
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (reverse (y :: ys)) = Some y
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (y :: ys) = Some y
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
y ∈ eval ρ base

            apply Hbase.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (x :: l) (tail (x :: l)))


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (x :: l) = Some m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
head (reverse (y :: ys)) = Some m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
last (y :: ys) = Some m

            apply Hlast.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
(λ new old : M, new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
(λ new old : M, new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
βˆ€ x : M,
  (λ old : M, x ∈ app_ext (eval ρ step) {[old]}) =
  flip
    (flip
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0: M
(λ old : M, x0 ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) x0

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0: M
βˆ€ x : M,
  (x0 ∈ app_ext (eval ρ step) {[x]}) =
  flip
    (flip
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) x0 x
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
x0, x1: M
(x0 ∈ app_ext (eval ρ step) {[x1]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) x0 x1

            reflexivity.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (x :: l) (tail (x :: l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (flip
           (Ξ» new old : M,
              new ∈ app_ext (eval ρ step) {[old]}))))
  (zip (x :: l) (tail (x :: l)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (flip
           (Ξ» new old : M,
              new ∈ app_ext (eval ρ step) {[old]}))))
  (zip (reverse (y :: ys)) (tail (reverse (y :: ys))))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (uncurry
     (flip
        (Ξ» new old : M,
           new ∈ app_ext (eval ρ step) {[old]})))
  (zip (y :: ys) (tail (y :: ys)))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x: M
l: list M
Feq: uncurry
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) =
(Ξ» x : M * M,
   let (old, new) := x in
   new ∈ app_ext (eval ρ step) {[old]})
y: M
ys: list M
Hlast: last (y :: ys) = Some m
Hbase: y ∈ eval ρ base
Hfa: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) ys)
Heq: x :: l = reverse (y :: ys)
Hfeq': (Ξ» new old : M,
   new ∈ app_ext (eval ρ step) {[old]}) =
flip
  (flip
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (y :: ys) (tail (y :: ys)))

          apply Hfa.
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
↔ is_witnessing_sequence m' (m' :: l) ∧ l β‰  []

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
↔ is_witnessing_sequence m' (m' :: l) ∧ l β‰  []

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
β†’ is_witnessing_sequence m' (m' :: l) ∧ l β‰  []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m' (m' :: l) ∧ l β‰  []
β†’ is_witnessing_sequence m l
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m l
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
β†’ is_witnessing_sequence m' (m' :: l) ∧ l β‰  []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
is_witnessing_sequence m' (m' :: l) ∧ l β‰  []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
is_witnessing_sequence m' (m' :: l)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
l β‰  []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
l β‰  []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head [] = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
[] β‰  []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
d :: l β‰  []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hhd: None = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
[] β‰  []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
d :: l β‰  []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', d: M
l: list M
lst: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head (d :: l) = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (d :: l) (tail (d :: l)))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
d :: l β‰  []
 discriminate. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hhd: head l = Some m
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
is_witnessing_sequence m' (m' :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
l: list M
lst: M
Hlst: last l = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l (tail l))
βˆ€ m : M,
  head l = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m' (m' :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: head [] = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m']
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: head (a :: l) = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: head [] = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m']
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hlst: last [] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
m: M
Hhd: None = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' [m']
 inversion Hhd.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: head (a :: l) = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: Some a = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: a :: l)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
a: M
l: list M
lst: M
Hlst: last (a :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l) (tail (a :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m: M
Hhd: Some a = Some m
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H0: a = m
is_witnessing_sequence m' (m' :: m :: l)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
l: list M
lst, m: M
Hhd: Some m = Some m
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: m :: l)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
l: list M
lst, m: M
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
is_witnessing_sequence m' (m' :: m :: l)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
l: list M
lst, m: M
Hlst: last (m :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l) (tail (m :: l)))
IHl: last l = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip l (tail l))
  β†’ βˆ€ m : M,
      head l = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last (m' :: m :: l) = Some lst ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: l) = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: l) (tail (m' :: m :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst, m: M
Hlst: last [m] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m]))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last (m' :: m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: d :: l) = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: d :: l)
         (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst, m: M
Hlst: last [m] = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m]))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst, m: M
Hlst: Some m = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last [m'; m] = Some lst ∧ lst ∈ eval ρ base)
∧ head [m'; m] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; m] (tail [m'; m]))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst, m: M
Hlst: Some m = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H0: m = lst
(βˆƒ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst]))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hlst: Some lst = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
(βˆƒ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst]))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
(βˆƒ lst0 : M,
   last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base)
∧ head [m'; lst] = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [m'; lst] (tail [m'; lst]))

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
βˆƒ lst0 : M,
  last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
head [m'; lst] = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [m'; lst] (tail [m'; lst]))

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
βˆƒ lst0 : M,
  last [m'; lst] = Some lst0 ∧ lst0 ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
last [m'; lst] = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Some lst = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Some lst = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 apply Hbase. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
head [m'; lst] = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [m'; lst] (tail [m'; lst]))

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Some m' = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    [(m', lst)]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Some m' = Some m'
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  [(m', lst)]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  [(m', lst)]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', lst)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]})) []

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', lst)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[lst]} ∧ m' ∈ app_interp M step' re
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
step' ∈ eval ρ step
∧ lst ∈ {[lst]} ∧ m' ∈ app_interp M step' lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]} ∧ m' ∈ app_interp M step' lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]} ∧ m' ∈ app_interp M step' lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
lst ∈ {[lst]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
m' ∈ app_interp M step' lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
m' ∈ app_interp M step' lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]})) []

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
lst: M
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [lst] (tail [lst]))
IHl: last [] = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip [] (tail []))
  β†’ βˆ€ m : M,
      head [] = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m' [m']
m', step': M
Hm': m' ∈ app_interp M step' lst
Hstep': step' ∈ eval ρ step
True
 exact I.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
(βˆƒ lst : M,
   last (m' :: m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m' :: m :: d :: l) = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m' :: m :: d :: l)
         (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
βˆƒ lst : M,
  last (m' :: m :: d :: l) = Some lst
  ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (m' :: m :: d :: l)
       (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
βˆƒ lst : M,
  last (m' :: m :: d :: l) = Some lst
  ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
last (m' :: m :: d :: l) = Some lst
∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
last (m' :: m :: d :: l) = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
last (d :: l) = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
last (d :: l) = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
lst ∈ eval ρ base
 apply Hbase. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (m' :: m :: d :: l)
       (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
head (m' :: m :: d :: l) = Some m'
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l) (tail (m :: d :: l)))
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l)))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m' :: m :: d :: l) (tail (m' :: m :: d :: l)))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m', m)
   :: (m, d)
      :: match l with
         | [] => []
         | x2 :: l2 => (d, x2) :: zip l l2
         end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (d :: l) (tail (d :: l)))
  β†’ βˆ€ m : M,
      head (d :: l) = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (m :: d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end
  β†’ βˆ€ m : M,
      Some d = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: last (d :: l) = Some lst
β†’ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end
  β†’ βˆ€ m : M,
      Some d = Some m
      β†’ βˆ€ m' step' : M,
          m' ∈ app_interp M step' m
          β†’ step' ∈ eval ρ step
            β†’ is_witnessing_sequence m'
                (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
β†’ βˆ€ m : M,
    Some d = Some m
    β†’ βˆ€ m' step' : M,
        m' ∈ app_interp M step' m
        β†’ step' ∈ eval ρ step
          β†’ is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
β†’ βˆ€ m : M,
    Some d = Some m
    β†’ βˆ€ m' step' : M,
        m' ∈ app_interp M step' m
        β†’ step' ∈ eval ρ step
          β†’ is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x = (m, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
Hwit: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
IHl: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
β†’ βˆ€ m : M,
    Some d = Some m
    β†’ βˆ€ m' step' : M,
        m' ∈ app_interp M step' m
        β†’ step' ∈ eval ρ step
          β†’ is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
IHl: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
β†’ βˆ€ m : M,
    Some d = Some m
    β†’ βˆ€ m' step' : M,
        m' ∈ app_interp M step' m
        β†’ step' ∈ eval ρ step
          β†’ is_witnessing_sequence m'
              (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hlst: last (d :: l) = Some lst
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    ((m, d)
     :: match l with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip l l2
        end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m', m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp M step' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
step' ∈ eval ρ step
∧ m ∈ {[m]} ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
m ∈ {[m]} ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
m ∈ {[m]} ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
m ∈ {[m]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
m' ∈ app_interp M step' m
 apply Hm'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d)
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    match l with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip l l2
    end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
uncurry
  (λ new old : M, new ∈ app_ext (eval ρ step) {[old]})
  (m, d)
 apply H1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m : M,
  Some d = Some m
  β†’ βˆ€ m' step' : M,
      m' ∈ app_interp M step' m
      β†’ step' ∈ eval ρ step
        β†’ is_witnessing_sequence m'
            (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m' step' : M,
  m' ∈ app_interp M step' d
  β†’ step' ∈ eval ρ step
    β†’ is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m' step' : M,
  m' ∈ app_interp M step' d
  β†’ step' ∈ eval ρ step
    β†’ is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'', d'': M
Hstep'': step'' ∈ eval ρ step
Hd'': d'' ∈ {[d]}
Hstep''d'': m ∈ app_interp M step'' d''
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m' step' : M,
  m' ∈ app_interp M step' d
  β†’ step' ∈ eval ρ step
    β†’ is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'', d'': M
Hstep'': step'' ∈ eval ρ step
Hd'': d'' ∈ {[d]}
Hstep''d'': m ∈ app_interp M step'' d''
H: d = d''
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d'', x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m' step' : M,
  m' ∈ app_interp M step' d
  β†’ step' ∈ eval ρ step
    β†’ is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Hd'': d ∈ {[d]}
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: βˆ€ m' step' : M,
  m' ∈ app_interp M step' d
  β†’ step' ∈ eval ρ step
    β†’ is_witnessing_sequence m' (m' :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: is_witnessing_sequence m (m :: d :: l)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: (βˆƒ lst : M,
   last (m :: d :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (m :: d :: l) = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (m :: d :: l) (tail (m :: d :: l)))
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
IHl: (βˆƒ lst : M,
   last (d :: l) = Some lst ∧ lst ∈ eval ρ base)
∧ Some m = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      ((m, d)
       :: match l with
          | [] => []
          | x2 :: l2 => (d, x2) :: zip l l2
          end)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x = (m, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
d: M
l: list M
lst, m: M
Hbase: lst ∈ eval ρ base
Hforall: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
m', step': M
Hm': m' ∈ app_interp M step' m
Hstep': step' ∈ eval ρ step
step'': M
Hstep'': step'' ∈ eval ρ step
Hstep''d'': m ∈ app_interp M step'' d
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, d)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
 apply H2.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
l: list M
is_witnessing_sequence m' (m' :: l) ∧ l β‰  []
β†’ is_witnessing_sequence m l
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

      Abort.


      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
↔ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
↔ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
β†’ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
β†’ is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
β†’ last (x :: l) = Some m
  ∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit: is_witnessing_sequence_old m (x :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last (x :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
is_witnessing_sequence_old m' ((x :: l) ++ [m'])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last (x :: l) = Some m
 apply Hwit1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
is_witnessing_sequence_old m' ((x :: l) ++ [m'])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last ((x :: l) ++ [m']) = Some m'
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
last ((x :: l) ++ [m']) = Some m'
 apply last_snoc. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
    end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ eval ρ base
 apply Hwit2. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit1: last (x :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ((d :: l) ++ [m']) l2
  end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(x, m')]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[x]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
True
 exact I. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[x]}

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[x]} ∧ m' ∈ app_interp M step' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
∧ x ∈ {[x]} ∧ m' ∈ app_interp M step' x
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ {[x]} ∧ m' ∈ app_interp M step' x

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
 apply Hstep'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ {[x]} ∧ m' ∈ app_interp M step' x

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ {[x]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' x

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x ∈ {[x]}
 constructor. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: last [x] = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' x

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: Some x = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' x
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit1: Some x = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [x] [])
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
H0: x = m
m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hwit2: m ∈ eval ρ base
Hwit1: Some m = Some m
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' m
 apply Hm'.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ((d :: l) ++ [m']) l2
  end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
      end)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
x0: (M * M)%type
l0: list (M * M)
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
H: x0 = (x, d)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d, x2) :: zip l l2
end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
Hwit3: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, d)
   :: match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end)
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
H1: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
    end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[x]}
 apply Hd. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hd: d ∈ app_ext (eval ρ step) {[x]}
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', d: M
l: list M
Hwit1: last (x :: d :: l) = Some m
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
  end


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip (l ++ [m']) l2
        end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
βˆ€ d : M,
  last [x; d] = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match [] ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip ([] ++ [m']) l2
        end
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
βˆ€ d : M,
  last (x :: d :: a :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      ((d, a) :: zip (a :: l) l)
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match (a :: l) ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip ((a :: l) ++ [m']) l2
        end

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
βˆ€ d : M,
  last [x; d] = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match [] ++ [m'] with
        | [] => []
        | x2 :: l2 => (d, x2) :: zip ([] ++ [m']) l2
        end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
D: M
Hm: last [x; D] = Some m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (D, x2) :: zip ([] ++ [m']) l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
D: M
Hm: last [x; D] = Some m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(D, m')]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
D: M
Hm: Some D = Some m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(D, m')]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
D: M
Hm: Some D = Some m
H0: D = m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Hm: Some m = Some m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')]

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) [(m, m')]

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[m]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[m]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
True
 exact I. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_ext (eval ρ step) {[m]}

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp M step' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
∧ m ∈ {[m]} ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m ∈ {[m]} ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
step' ∈ eval ρ step
 apply Hstep'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m ∈ {[m]} ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m ∈ {[m]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m ∈ {[m]}
 constructor. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
m' ∈ app_interp M step' m

            apply Hm'.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
βˆ€ d : M,
  last (x :: d :: a :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      ((d, a) :: zip (a :: l) l)
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match (a :: l) ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip ((a :: l) ++ [m']) l2
        end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (a :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 => (d, x2) :: zip ((a :: l) ++ [m']) l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
x0: (M * M)%type
l0: list (M * M)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
H: x0 = (d, a)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (a, x2) :: zip l l2
end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a)
   :: match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
      end)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
    end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
a ∈ app_ext (eval ρ step) {[d]}
 apply H1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip (l ++ [m']) l2
  end

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (x :: a :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (x :: a :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
last (a :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m', a: M
l: list M
Hwit2: x ∈ eval ρ base
step': M
Hstep': step' ∈ eval ρ step
Hm': m' ∈ app_interp M step' m
IHl: βˆ€ d : M,
  last (x :: d :: l) = Some m
  β†’ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      match l with
      | [] => []
      | x2 :: l2 => (d, x2) :: zip l l2
      end
    β†’ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        match l ++ [m'] with
        | [] => []
        | x2 :: l2 =>
            (d, x2) :: zip (l ++ [m']) l2
        end
d: M
Hlast: last (x :: d :: a :: l) = Some m
Hwit: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, a) :: zip (a :: l) l)
H1: a ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l l2
  end

            apply H2.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
last (x :: l) = Some m
∧ is_witnessing_sequence_old m' ((x :: l) ++ [m'])
β†’ is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: is_witnessing_sequence_old m' ((x :: l) ++ [m'])
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: is_witnessing_sequence_old m' ((x :: l) ++ [m'])
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Heqlen: len = length l
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Heqlen: len = length l
length l ≀ len
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Heqlen: len = length l
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Heqlen: len = length l
length l ≀ len
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Heqlen: len = length l
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: match (x :: l) ++ [m'] with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
len: nat
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
H: x ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l ++ [m'] with
    | [] => []
    | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
    end
len: nat
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
is_witnessing_sequence_old m (x :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
last (x :: l) = Some m
∧ (x ∈ eval ρ base
   ∧ Forall
       (Ξ» x : M * M,
          let (old, new) := x in
          new ∈ app_ext (eval ρ step) {[old]})
       (zip (x :: l) l))
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
last (x :: l) = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
(x ∈ eval ρ base
 ∧ Forall
     (Ξ» x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     (zip (x :: l) l))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
last (x :: l) = Some m
 apply Hlast. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
(x ∈ eval ρ base
 ∧ Forall
     (Ξ» x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     (zip (x :: l) l))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
x ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (x :: l) l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
x ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
x ∈ eval ρ base
 apply Hbase. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hbase: x ∈ eval ρ base
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, x, m': M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
len: nat
Hlen: length l ≀ len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ 0
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ S len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ 0
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
Hlen: length l ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: length (d :: l) ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: length (d :: l) ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x, d: M
l: list M
Hlast: last (x :: d :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (d :: l) ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((d :: l) ++ [m']) l2
  end
Hlen: S (length l) ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: d :: l) (d :: l))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 lia.  
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
True
 exact I. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
x0: (M * M)%type
l: list (M * M)
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x0 = (x, m')
H0: l = []
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(x, m')]
Hlen: length [] ≀ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≀ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≀ 0
H1: m' ∈ app_ext (eval ρ step) {[x]}
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≀ 0
step', m'': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[x]}
Hstep'm'': m' ∈ app_interp M step' m''
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', x: M
Hlast: last [x] = Some m
Hlen: length [] ≀ 0
step', m'': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[x]}
Hstep'm'': m' ∈ app_interp M step' m''
H: x = m''
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hm'm'': m'' ∈ {[m'']}
Hstep'm'': m' ∈ app_interp M step' m''
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: last [m''] = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: Some m'' = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m', m'': M
Hlast: Some m'' = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m''
H0: m'' = m
m' ∈ app_interp M step' m
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
Hlast: Some m = Some m
Hlen: length [] ≀ 0
step': M
Hstep': step' ∈ eval ρ step
Hstep'm'': m' ∈ app_interp M step' m
m' ∈ app_interp M step' m

            apply Hstep'm''.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ S len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
l: list M
Hlast: last (x :: l) = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip (l ++ [m']) l2
  end
Hlen: length l ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: l) l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (x' :: l') ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((x' :: l') ++ [m']) l2
  end
Hlen: length (x' :: l') ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last [x] = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
Hlast: Some x = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [x] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
Hlast: Some x = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
H0: x = m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x: M
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (x, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
H0: x = m
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match [] ++ [m'] with
  | [] => []
  | x2 :: l2 => (m, x2) :: zip ([] ++ [m']) l2
  end
Hlen: length [] ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≀ S len
x: (M * M)%type
l: list (M * M)
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x = (m, m')
H0: l = []
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(m, m')]
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [m] [])
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
True
 exact I. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
H1: m' ∈ app_ext (eval ρ step) {[m]}
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step': M
Hstep': βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp M step' re
βˆƒ step' : M,
  step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step': M
Hstep': βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ {[m]} ∧ m' ∈ app_interp M step' re
step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
step' ∈ eval ρ step
 apply Hstep'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
Hlen: length [] ≀ S len
step', m'': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m]}
Hm'': m' ∈ app_interp M step' m''
H: m = m''
m' ∈ app_interp M step' m''
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m': M
len: nat
m'': M
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m''
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m'')
Hlen: length [] ≀ S len
step': M
Hstep': step' ∈ eval ρ step
Hmm'': m'' ∈ {[m'']}
Hm'': m' ∈ app_interp M step' m''
m' ∈ app_interp M step' m''

              apply Hm''.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match (x' :: l') ++ [m'] with
  | [] => []
  | x2 :: l2 =>
      (x, x2) :: zip ((x' :: l') ++ [m']) l2
  end
Hlen: length (x' :: l') ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          (zip (x :: l) l)
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x :: x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≀ S len
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≀ S len
x0: (M * M)%type
l: list (M * M)
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
H: x0 = (x, x')
H0: l =
match l' ++ [m'] with
| [] => []
| x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hall: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x', x2) :: zip (l' ++ [m']) l2
      end)
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
IHlen: βˆ€ (x : M) (l : list M),
  last (x :: l) = Some m
  β†’ Forall
      (Ξ» x0 : M * M,
         let (old, new) := x0 in
         new ∈ app_ext (eval ρ step) {[old]})
      match l ++ [m'] with
      | [] => []
      | x2 :: l2 =>
          (x, x2) :: zip (l ++ [m']) l2
      end
    β†’ length l ≀ len
      β†’ Forall
          (Ξ» x0 : M * M,
             let (old, new) := x0 in
             new
             ∈ app_ext (eval ρ step) {[old]})
          match l with
          | [] => []
          | x2 :: l2 => (x, x2) :: zip l l2
          end
        ∧ (βˆƒ step' : M,
             step' ∈ eval ρ step
             ∧ m' ∈ app_interp M step' m)
x, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (x :: x' :: l') (x' :: l'))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((x, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
(x' ∈ app_ext (eval ρ step) {[x]}
 ∧ Forall
     (Ξ» x : M * M,
        let (old, new) := x in
        new ∈ app_ext (eval ρ step) {[old]})
     match l' with
     | [] => []
     | x2 :: l2 => (x', x2) :: zip l' l2
     end)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]}
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
x' ∈ app_ext (eval ρ step) {[x]}
 apply H1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step ∧ m' ∈ app_interp M step' m)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: length (x' :: l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
length l' ≀ len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, m': M
len: nat
x': M
l': list M
IHlen: length l' ≀ len
β†’ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    match l' with
    | [] => []
    | x2 :: l2 => (x', x2) :: zip l' l2
    end
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
x: M
Hlast: last (x' :: l') = Some m
Hlen: S (length l') ≀ S len
H1: x' ∈ app_ext (eval ρ step) {[x]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' ++ [m'] with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip (l' ++ [m']) l2
  end
length l' ≀ len
 lia.
      Qed.

      Definition witnessed_elements_old : propset (Domain M) :=
        PropSet (Ξ» m, βˆƒ l, is_witnessing_sequence_old m l).

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
F witnessed_elements_old βŠ† witnessed_elements_old

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
F witnessed_elements_old βŠ† witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
βˆ€ x : M,
  x ∈ F witnessed_elements_old
  β†’ x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x ∈ F witnessed_elements_old
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ Fassoc ρ
    patt_ind_gen_body^{svar:0↦
    fresh_svar patt_ind_gen_body}
    (fresh_svar patt_ind_gen_body)
    witnessed_elements_old
x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    patt_ind_gen_body^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    patt_ind_gen_body^[svar:0↦
    patt_free_svar (fresh_svar patt_ind_gen_body)]
x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (((nest_mu base)^[svar:0↦
      patt_free_svar
        (fresh_svar patt_ind_gen_body)] ---> βŠ₯) --->
     (nest_mu step)^[svar:0↦
     patt_free_svar
       (fresh_svar patt_ind_gen_body)] $
     patt_free_svar
       (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^[svar:0↦
    patt_free_svar (fresh_svar patt_ind_gen_body)]
  βˆͺ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      ((nest_mu step)^[svar:0↦
       patt_free_svar
         (fresh_svar patt_ind_gen_body)] $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
  βˆͺ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      ((nest_mu step)^[svar:0↦
       patt_free_svar
         (fresh_svar patt_ind_gen_body)] $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
Hx: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
  βˆͺ eval
      (update_svar_val
         (fresh_svar patt_ind_gen_body)
         witnessed_elements_old ρ)
      ((nest_mu step)^{svar:0↦
       fresh_svar patt_ind_gen_body} $
       patt_free_svar
         (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ witnessed_elements_old
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    ((nest_mu step)^{svar:0↦
     fresh_svar patt_ind_gen_body} $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x
∈ {[ m | βˆƒ l : list M, is_witnessing_sequence_old m l ]}

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
is_witnessing_sequence_old x [x]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
last [x] = Some x
∧ x ∈ eval ρ base
  ∧ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]})
      (zip [x] [])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
Some x = Some x
∧ x ∈ eval ρ base
  ∧ Forall
      (Ξ» x : M * M,
         let (old, new) := x in
         new ∈ app_ext (eval ρ step) {[old]}) []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
Some x = Some x
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
Some x = Some x
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]}) []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
 constructor. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  (nest_mu base)^{svar:0↦fresh_svar patt_ind_gen_body}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval ρ
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body)
  (nest_mu base)^{svar:0↦fresh_svar patt_ind_gen_body}

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
free_svars
  (nest_mu base)^{svar:0↦fresh_svar patt_ind_gen_body}
βŠ† free_svars ?Ο•β‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 auto. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
free_svars
  (nest_mu base)^{svar:0↦fresh_svar patt_ind_gen_body}
βŠ† free_svars patt_ind_gen_body

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
free_svars
  (nest_mu base)^[svar:0↦patt_free_svar
                           (fresh_svar
                              patt_ind_gen_body)]
βŠ† free_svars patt_ind_gen_body

            solve_free_svars_inclusion 5.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval ρ
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval ρ
    (nest_mu base)^{svar:0↦
    fresh_svar patt_ind_gen_body}
x ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval ρ
    (nest_mu base)^[svar:0↦
    patt_free_svar (fresh_svar patt_ind_gen_body)]
x ∈ eval ρ base

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x ∈ eval ρ base
x ∈ eval ρ base
 auto.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    ((nest_mu step)^{svar:0↦
     fresh_svar patt_ind_gen_body} $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    ((nest_mu step)^{svar:0↦
     fresh_svar patt_ind_gen_body} $
     patt_free_svar (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (nest_mu step)^{svar:0↦
       fresh_svar patt_ind_gen_body})
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (patt_free_svar
          (fresh_svar patt_ind_gen_body)))
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (nest_mu step)^{svar:0↦
       fresh_svar patt_ind_gen_body})
    (svar_valuation
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (fresh_svar patt_ind_gen_body))
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ app_ext
    (eval
       (update_svar_val
          (fresh_svar patt_ind_gen_body)
          witnessed_elements_old ρ)
       (nest_mu step)^{svar:0↦
       fresh_svar patt_ind_gen_body})
    witnessed_elements_old
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ {[ e | βˆƒ le re : M,
           le
           ∈ eval
               (update_svar_val
                  (fresh_svar patt_ind_gen_body)
                  witnessed_elements_old ρ)
               (nest_mu step)^{svar:0↦
               fresh_svar patt_ind_gen_body}
           ∧ re ∈ witnessed_elements_old
             ∧ e ∈ app_interp M le re ]}
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
H2: m ∈ witnessed_elements_old
Happ: x ∈ app_interp M step' m
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
H2: m
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l ]}
Happ: x ∈ app_interp M step' m
x ∈ witnessed_elements_old

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp M step' m
x ∈ witnessed_elements_old


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp M step' m
x
∈ {[ m | βˆƒ l : list M, is_witnessing_sequence_old m l ]}

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Hl: is_witnessing_sequence_old m l
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x (l ++ [x])


          (* `l` is not empty *)
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ([] ++ [x])
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ([] ++ [x])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Heql: l = []
Hl: is_witnessing_sequence_old m []
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ([] ++ [x])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Heql: l = []
H: last [] = Some m
H0: False
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ([] ++ [x])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
Heql: l = []
H: None = Some m
H0: False
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ([] ++ [x])
 inversion H. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^{svar:0↦
    fresh_svar patt_ind_gen_body}
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ)
    (nest_mu step)^[svar:0↦
    patt_free_svar (fresh_svar patt_ind_gen_body)]
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step'
∈ eval
    (update_svar_val
       (fresh_svar patt_ind_gen_body)
       witnessed_elements_old ρ) step
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step' ∈ eval ρ step
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) step

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
free_svars step βŠ† free_svars ?Ο•β‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
svar_is_fresh_in (fresh_svar patt_ind_gen_body) ?Ο•β‚‚
 apply set_svar_fresh_is_fresh. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
free_svars step βŠ† free_svars patt_ind_gen_body

            solve_free_svars_inclusion 2.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step' ∈ eval ρ step
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step' ∈ eval ρ step
P: is_witnessing_sequence_old ?m (?x :: ?l)
∧ (βˆƒ step'0 : M,
     step'0 ∈ eval ρ step
     ∧ ?m' ∈ app_interp M step'0 ?m)
↔ last (?x :: ?l) = Some ?m
  ∧ is_witnessing_sequence_old 
      ?m' ((?x :: ?l) ++ [?m'])
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step' ∈ eval ρ step
P: is_witnessing_sequence_old ?m (?x :: ?l)
∧ (βˆƒ step'0 : M,
     step'0 ∈ eval ρ step
     ∧ ?m' ∈ app_interp M step'0 ?m)
β†’ last (?x :: ?l) = Some ?m
  ∧ is_witnessing_sequence_old 
      ?m' ((?x :: ?l) ++ [?m'])
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x, step', m: M
l: list M
mβ‚€: M
l': list M
Heql: l = mβ‚€ :: l'
Hl: is_witnessing_sequence_old m (mβ‚€ :: l')
Happ: x ∈ app_interp M step' m
H1: step' ∈ eval ρ step
P: last (mβ‚€ :: l') = Some m
∧ is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])
is_witnessing_sequence_old x ((mβ‚€ :: l') ++ [x])

          apply P.
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen βŠ† witnessed_elements_old

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen βŠ† witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
well_formed_positive (mu , patt_ind_gen_body)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Fassoc ρ
  patt_ind_gen_body^{svar:0↦
  fresh_svar patt_ind_gen_body}
  (fresh_svar patt_ind_gen_body)
  witnessed_elements_old βŠ† witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
well_formed_positive (mu , patt_ind_gen_body)
 apply patt_ind_gen_wfp. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Fassoc ρ
  patt_ind_gen_body^{svar:0↦fresh_svar
                              patt_ind_gen_body}
  (fresh_svar patt_ind_gen_body)
  witnessed_elements_old βŠ† witnessed_elements_old

        apply witnessed_elements_old_prefixpoint.
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
let Sfix := eval ρ patt_ind_gen in F Sfix = Sfix

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
let Sfix := eval ρ patt_ind_gen in F Sfix = Sfix

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
well_formed_positive (mu , patt_ind_gen_body)

        apply patt_ind_gen_wfp.
      Qed.


      Definition witnessed_elements_old_of_max_len len : propset (Domain M) :=
        PropSet (Ξ» m, βˆƒ l, is_witnessing_sequence_old m l /\ length l <= len).

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old_of_max_len 1
βŠ† eval ρ patt_ind_gen
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
βŠ† eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old_of_max_len 1
βŠ† eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old_of_max_len 1
βŠ† eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
β†’ m ∈ eval ρ patt_ind_gen

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
β†’ m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
m ∈ witnessed_elements_old_of_max_len 1
β†’ m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
H: m ∈ witnessed_elements_old_of_max_len 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
Hlast: last l = Some m
Hm: match l with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
Hlen: length l ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlast: last [] = Some m
Hm: False
Hlen: length [] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlast: last [] = Some m
Hm: False
Hlen: length [] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlast: None = Some m
Hm: False
Hlen: length [] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 inversion Hlast. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
Hlast: last [d] = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: length (d :: d0 :: l) ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: length (d :: d0 :: l) ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (S (length l)) ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
Hlast: last [d] = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
Hlast: Some d = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≀ 1
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
Hlast: Some d = Some m
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≀ 1
H0: d = m
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m, d: M
Hm: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])
Hlen: length [d] ≀ 1
H0: d = m
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlen: length [m] ≀ 1
Hm: m ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [m] [])
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlen: length [m] ≀ 1
Hm: m ∈ eval ρ base
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlen: length [m] ≀ 1
Hm: m ∈ eval ρ base
m ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
Hlen: length [m] ≀ 1
Hm: m ∈ eval ρ base
m ∈ eval ρ base
 apply Hm.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
βŠ† eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
witnessed_elements_old_of_max_len (S (S len))
βŠ† eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
β†’ m ∈ eval ρ patt_ind_gen

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
β†’ m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
m ∈ witnessed_elements_old_of_max_len (S (S len))
β†’ m ∈ F (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
H: m ∈ witnessed_elements_old_of_max_len (S (S len))
m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
m ∈ F (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
m ∈ F (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hwit: is_witnessing_sequence_old m l
Hlen: length l ≀ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
l: list M
Hlast: last l = Some m
Hl: match l with
| [] => False
| mβ‚€ :: ms =>
    mβ‚€ ∈ eval ρ base
    ∧ Forall
        (Ξ» x : M * M,
           let (old, new) := x in
           new ∈ app_ext (eval ρ step) {[old]})
        (zip (mβ‚€ :: ms) ms)
end
Hlen: length l ≀ S (S len)
Hwit': is_witnessing_sequence_old m l
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
Hlast: last [] = Some m
Hl: False
Hlen: length [] ≀ S (S len)
Hwit': is_witnessing_sequence_old m []
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
Hlast: last [] = Some m
Hl: False
Hlen: length [] ≀ S (S len)
Hwit': is_witnessing_sequence_old m []
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 contradiction. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: length (d :: l) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hl: d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip (d :: l) l)
Hlen: S (length l) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
l: list M
Hlast: last (d :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: l) l)
Hlen: S (length l) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: last [d] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: last [d] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some d = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d]
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some d = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d]
H0: d = m
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d] [])
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d]
H0: d = m
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m: M
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [m] [])
Hd: m ∈ eval ρ base
Hlen: S (length []) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [m]
m ∈ eval ρ base
 apply Hd.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m
∈ eval ρ base
  βˆͺ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
l: list M
Hlast: last (d :: d0 :: l) = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: l) (d0 :: l))
Hlen: S (length (d0 :: l)) ≀ S (S len)
Hwit': is_witnessing_sequence_old m (d :: d0 :: l)
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
Hlast: last [d; d0] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
Hlast: last [d; d0] = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
Hlast: Some d0 = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0: M
Hlast: Some d0 = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; d0] [d0])
Hlen: S (length [d0]) ≀ S (S len)
Hwit': is_witnessing_sequence_old m [d; d0]
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
H0: d0 = m
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: is_witnessing_sequence_old d [d]
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' d)
↔ last [d] = Some d
  ∧ is_witnessing_sequence_old m ([d] ++ [m])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: last [d] = Some d
∧ is_witnessing_sequence_old m ([d] ++ [m])
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip [d; m] [m])
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
x: (M * M)%type
l: list (M * M)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
H: x = (d, m)
H0: l = []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  [(d, m)]
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Some d = Some d
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Some d = Some d
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: Some d = Some d
∧ is_witnessing_sequence_old m [d; m]
β†’ is_witnessing_sequence_old d [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
P: is_witnessing_sequence_old d [d]
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' d)
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
∧ m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
∧ m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ eval ρ patt_ind_gen
    ∧ m ∈ app_interp M step' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
∧ m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
∧ d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp M step' d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
∧ d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp M step' d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp M step' d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
step' ∈ eval ρ step
 apply Hstep'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen ∧ m ∈ app_interp M step' d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_interp M step' d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ witnessed_elements_old_of_max_len (S len)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d] ∧ length [d] ≀ S len

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
length [d] ≀ S len

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
length [d] ≀ S len
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
1 ≀ S len
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
is_witnessing_sequence_old d [d]

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
last [d] = Some d
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
last [d] = Some d
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
∧ Forall
    (Ξ» x : M * M,
       let (old, new) := x in
       new ∈ app_ext (eval ρ step) {[old]})
    (zip [d] [])

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) 
  (zip [d] [])

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
d ∈ eval ρ base
 apply Hd. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) (zip [d] [])

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
True

                exact I.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d: M
Hlast: Some m = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; m]
Hlen: S (length [m]) ≀ S (S len)
Hwitd: is_witnessing_sequence_old d [d]
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' d
H1: m ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]}) []
Hsomed: Some d = Some d
m ∈ app_interp M step' d

              apply Hm.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        ((x :: l) ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (d :: d0 :: x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  (zip (d :: d0 :: x' :: l') (d0 :: x' :: l'))
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
x: (M * M)%type
l: list (M * M)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
H: x = (d, d0)
H0: l =
(d0, x')
:: match l' with
   | [] => []
   | x2 :: l2 => (x', x2) :: zip l' l2
   end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hl: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d, d0)
   :: (d0, x')
      :: match l' with
         | [] => []
         | x2 :: l2 => (x', x2) :: zip l' l2
         end)
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
x: (M * M)%type
l: list (M * M)
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
H: x = (d0, x')
H0: l =
match l' with
| [] => []
| x2 :: l2 => (x', x2) :: zip l' l2
end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H2: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  ((d0, x')
   :: match l' with
      | [] => []
      | x2 :: l2 => (x', x2) :: zip l' l2
      end)
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= rev (d0 :: x' :: l') !! 1: option M
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Inhabited M
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l'))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Inhabited M
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
M
 exact d0. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l'))

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev (d0 :: x' :: l'))
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length ((rev l' ++ [x']) ++ [d0])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l' ++ [x']) + length [d0]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l') + length [x'] + length [d0]
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
1 < length (rev l') + 1 + 1
  lia.  
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Htot: rev (d0 :: x' :: l') !! 1 =
Some (rev (d0 :: x' :: l') !!! 1)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
IHlen: witnessed_elements_old_of_max_len (S len)
βŠ† eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
mprev: M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ witnessed_elements_old_of_max_len (S len)
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: βˆ€ (m x m' : M) (l : list M),
  is_witnessing_sequence_old m (x :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m' ∈ app_interp M step' m)
  ↔ last (x :: l) = Some m
    ∧ is_witnessing_sequence_old m'
        (x :: l ++ [m'])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d0 :: x' :: l' =
rev (tail (rev (d0 :: x' :: l'))) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
[x'] = rev [x']
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
[x'] = rev [x']
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ [x']) ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail ((rev l' ++ rev [x']) ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
[d0] = rev [d0]
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
[d0] = rev [d0]
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([x'] ++ l') ++ rev [d0])) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' =
rev (tail (rev ([d0] ++ [x'] ++ l'))) ++ [m]

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
last ([d0] ++ [x'] ++ l') = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
last (x' :: l') = Some m
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Hx'r: [x'] = rev [x']
Hd0r: [d0] = rev [d0]
d0 :: x' :: l' = [d0] ++ [x'] ++ l'

              reflexivity.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: is_witnessing_sequence_old mprev
  (d :: rev (tail (rev (d0 :: x' :: l'))))
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
↔ last (d :: rev (tail (rev (d0 :: x' :: l')))) =
  Some mprev
  ∧ is_witnessing_sequence_old m
      (d
       :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last (d :: rev (tail (rev (d0 :: x' :: l')))) =
Some mprev
∧ is_witnessing_sequence_old m
    (d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m])
β†’ is_witnessing_sequence_old mprev
    (d :: rev (tail (rev (d0 :: x' :: l'))))
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' =
d :: rev (tail (rev (d0 :: x' :: l'))) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)



            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = []
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: (d1 :: l) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = []
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
Heqrev: length (rev (tail (rev (d0 :: x' :: l')))) =
0
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (tail (rev (d0 :: x' :: l'))) = 0
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (rev (d0 :: x' :: l')) = 0
∨ length (rev (d0 :: x' :: l')) = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: length (d0 :: x' :: l') = 0
∨ length (d0 :: x' :: l') = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
P: last [d] = Some mprev
∧ is_witnessing_sequence_old m (d :: [] ++ [m])
β†’ is_witnessing_sequence_old mprev [d]
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: [] ++ [m]
Heqrev: S (S (length l')) = 0 ∨ S (S (length l')) = 1
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 lia.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: (d1 :: l) ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: d1 :: l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
Hlen: S (length (d0 :: x' :: l')) ≀ S (S len)
Hwit': is_witnessing_sequence_old m
  (d :: d0 :: x' :: l')
d1: M
l: list M
Heqrev: rev (tail (rev (d0 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H1: d0 ∈ app_ext (eval ρ step) {[d]}
H3: x' ∈ app_ext (eval ρ step) {[d0]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
mp:= ((rev l' ++ [x']) ++ [d0]) !! 1: option M
Heqmprev: mprev = rev (d0 :: x' :: l') !!! 1
Htot: rev (d0 :: x' :: l') !! 1 = Some mprev
Heq1: d :: d0 :: x' :: l' = d :: d1 :: l ++ [m]
H0: d0 = d1
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Heq1: d :: d1 :: x' :: l' = d :: d1 :: l ++ [m]
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d :: d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: (d1 :: l) ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: l ++ [m])
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = l ++ [m]
match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [] ++ [m]
Some d1 = Some mprev
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [] ++ [m]
Some d1 = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [m]
Some d1 = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = [m]
H0: x' = m
H1: l' = []
Some d1 = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
d1, m: M
IHlen: rev [d1; m] !!! 1
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ rev [d1; m] !!! 1 ∈ eval ρ patt_ind_gen
d: M
Hlast: last [m] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; m]
Hlen: S (length [d1; m]) ≀ S (S len)
Htot: rev [d1; m] !! 1 = Some (rev [d1; m] !!! 1)
H2: [m] = [m]
Some d1 = Some (rev [d1; m] !!! 1)
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = (d0 :: l) ++ [m]
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = d0 :: l ++ [m]
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
d0: M
l: list M
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
H2: x' :: l' = d0 :: l ++ [m]
H0: x' = d0
H1: l' = l ++ [m]
last (d0 :: l) = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: rev (d1 :: d0 :: l ++ [m]) !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
H2: d0 :: l ++ [m] = d0 :: l ++ [m]
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≀ S (S len)
last (d0 :: l) = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: rev (d1 :: d0 :: l ++ [m]) !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≀ S (S len)
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Htot: ((rev (l ++ [m]) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: l ++ [m]) !!! 1
Hlast: last (d0 :: l ++ [m]) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: l ++ [m])
Hlen: S (length (d1 :: d0 :: l ++ [m])) ≀ S (S len)
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≀ S (S len)
last (d0 :: l) = Some mprev
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Htot: ((rev (d2 :: l0) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≀ S (S len)
last (d0 :: l) = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≀ S (S len)
Hcontra': length (l ++ [m]) = length l + length [m]
last (d0 :: l) = Some mprev

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≀ S (S len)
Hcontra': length [] = length l + length [m]
last (d0 :: l) = Some mprev

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1: M
Hcontra: l ++ [m] = []
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 1 = Some mprev
Heqmprev: mprev = rev [d1; d0] !!! 1
Hlast: last [d0] = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m [d; d1; d0]
Hlen: S (length [d1; d0]) ≀ S (S len)
Hcontra': 0 = length l + 1
last (d0 :: l) = Some mprev
 lia.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Htot: ((rev (d2 :: l0) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev (l ++ [m]) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
last (d0 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev [m] ++ rev l) ++ [d0]) ++ [d1]) !! 1 =
Some mprev
last (d0 :: l) = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0: M
l: list M
d1, d2: M
l0: list M
Hcontra: l ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev l ++ [d0]) ++ [d1]) !! 0 = Some mprev
last (d0 :: l) = Some mprev


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 0 = Some mprev
last [d0] = Some mprev
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev (d3 :: l) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev [] ++ [d0]) ++ [d1]) !! 0 = Some mprev
last [d0] = Some mprev
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d1, d2: M
l0: list M
Hcontra: [] ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: Some d0 = Some mprev
last [d0] = Some mprev
 apply Htot. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: ((rev (d3 :: l) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = Some mprev

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) =
(((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) =
head (((rev l ++ [d3]) ++ [d0]) ++ [d1])

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = head ((rev l ++ [d3]) ++ [d0])
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d0 :: d3 :: l) = head (rev l ++ [d3])

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, d0, d3: M
l: list M
d1, d2: M
l0: list M
Hcontra: (d3 :: l) ++ [m] = d2 :: l0
Heqmprev: mprev = rev (d1 :: d0 :: d2 :: l0) !!! 1
Hlast: last (d0 :: d2 :: l0) = Some m
Hd: d ∈ eval ρ base
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: d0 :: d2 :: l0)
Hlen: S (length (d1 :: d0 :: d2 :: l0)) ≀ S (S len)
Htot: (((rev l ++ [d3]) ++ [d0]) ++ [d1]) !! 0 =
Some mprev
last (d3 :: l) = head (rev l ++ [d3])

              apply last_rev_head.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: last (d1 :: l) = Some mprev
∧ is_witnessing_sequence_old m
    (d :: d1 :: x' :: l')
β†’ is_witnessing_sequence_old mprev (d :: d1 :: l)
  ∧ (βˆƒ step' : M,
       step' ∈ eval ρ step
       ∧ m ∈ app_interp M step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hwit': is_witnessing_sequence_old m
  (d :: d1 :: x' :: l')
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
Hm: match l with
| [] => Some d1
| _ :: _ => last l
end = Some mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
P: is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ (βˆƒ step' : M,
     step' ∈ eval ρ step
     ∧ m ∈ app_interp M step' mprev)
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
m ∈ app_ext (eval ρ step) (eval ρ patt_ind_gen)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
βˆƒ re : M,
  step' ∈ eval ρ step
  ∧ re ∈ eval ρ patt_ind_gen
    ∧ m ∈ app_interp M step' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
step' ∈ eval ρ step
∧ mprev ∈ eval ρ patt_ind_gen
  ∧ m ∈ app_interp M step' mprev

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
step' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
mprev ∈ eval ρ patt_ind_gen
∧ m ∈ app_interp M step' mprev

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
step' ∈ eval ρ step
 apply Hstep'. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
mprev ∈ eval ρ patt_ind_gen
∧ m ∈ app_interp M step' mprev

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
mprev ∈ eval ρ patt_ind_gen
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
m ∈ app_interp M step' mprev

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
m ∈ app_interp M step' mprev
 apply Hm. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
mprev ∈ eval ρ patt_ind_gen

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l)
∧ length (d :: d1 :: l) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
length (d :: d1 :: l) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
is_witnessing_sequence_old mprev (d :: d1 :: l)
 apply Hwitmprev. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
length (d :: d1 :: l) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (length (d1 :: x' :: l')) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
S (S (length l)) ≀ S len
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
S (S (length l)) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
S (length l') ≀ len
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
S (S (length l)) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
S (length l') ≀ len
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
S (S (length l)) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
length l' = length l
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
H: length l' = length l
S (S (length l)) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
length l' = length l
 apply (@list_len_slice _ _ _ _ _ H2). 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
H: length l' = length l
S (S (length l)) ≀ S len

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
len: nat
mprev: M
IHlen: mprev
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l
           ∧ length l ≀ S len ]}
β†’ mprev ∈ eval ρ patt_ind_gen
m, d, x': M
l': list M
Hlast: last (x' :: l') = Some m
Hd: d ∈ eval ρ base
d1: M
Hlen: S (S (S (length l'))) ≀ S (S len)
l: list M
Heqrev: rev (tail (rev (d1 :: x' :: l'))) = d1 :: l
H3: x' ∈ app_ext (eval ρ step) {[d1]}
H1: d1 ∈ app_ext (eval ρ step) {[d]}
H4: Forall
  (Ξ» x : M * M,
     let (old, new) := x in
     new ∈ app_ext (eval ρ step) {[old]})
  match l' with
  | [] => []
  | x2 :: l2 => (x', x2) :: zip l' l2
  end
Htot: rev (d1 :: x' :: l') !! 1 = Some mprev
Heqmprev: mprev = rev (d1 :: x' :: l') !!! 1
mp:= ((rev l' ++ [x']) ++ [d1]) !! 1: option M
H2: x' :: l' = l ++ [m]
Hwitmprev: is_witnessing_sequence_old mprev
  (d :: d1 :: l)
step': M
Hstep': step' ∈ eval ρ step
Hm: m ∈ app_interp M step' mprev
Hlen': S (length l') ≀ len
H: length l' = length l
S (S (length l')) ≀ S len
 lia.
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old βŠ† eval ρ patt_ind_gen

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old βŠ† eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x ∈ witnessed_elements_old
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x ∈ witnessed_elements_old
x ∈ eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
H: x
∈ {[ m | βˆƒ l : list M,
           is_witnessing_sequence_old m l ]}
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
H': x
∈ witnessed_elements_old_of_max_len (length wit)
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
x ∈ witnessed_elements_old_of_max_len (length wit)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
is_witnessing_sequence_old x wit
∧ length wit ≀ length wit
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
is_witnessing_sequence_old x wit
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
length wit ≀ length wit
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
length wit ≀ length wit
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hwit: is_witnessing_sequence_old x wit
H': x
∈ witnessed_elements_old_of_max_len (length wit)
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hl: wit = []
Hwit: is_witnessing_sequence_old x []
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
Hl: wit = []
H: last [] = Some x
H0: False
H': x ∈ witnessed_elements_old_of_max_len (length [])
x ∈ eval ρ patt_ind_gen
 contradiction. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (length (y :: l'))
x ∈ witnessed_elements_old_of_max_len (S ?len)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
x: M
wit: list M
y: M
l': list M
Hl: wit = y :: l'
Hwit: is_witnessing_sequence_old x (y :: l')
H': x
∈ witnessed_elements_old_of_max_len
    (S (length l'))
x ∈ witnessed_elements_old_of_max_len (S ?len)

        apply H'.
      Qed.

      Definition witnessed_elements : propset (Domain M) :=
        PropSet (Ξ» m, βˆƒ l, is_witnessing_sequence m l).

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old = witnessed_elements

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old = witnessed_elements

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old βŠ† witnessed_elements
βŠ† witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
(βˆ€ x : M,
   x ∈ witnessed_elements_old β†’ x ∈ witnessed_elements)
∧ (βˆ€ x : M,
     x ∈ witnessed_elements
     β†’ x ∈ witnessed_elements_old)

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
m ∈ {[ m | βˆƒ l : list M, is_witnessing_sequence m l ]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence m l
m
∈ {[ m | βˆƒ l : list M, is_witnessing_sequence_old m l ]}

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
m ∈ {[ m | βˆƒ l : list M, is_witnessing_sequence m l ]}
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence m (reverse l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence_old m (reverse (reverse l))

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence_old m l
is_witnessing_sequence_old m l

          exact H.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence m l
m
∈ {[ m | βˆƒ l : list M, is_witnessing_sequence_old m l ]}
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence m l
is_witnessing_sequence_old m (reverse l)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
m: M
l: list M
H: is_witnessing_sequence m l
is_witnessing_sequence m l

          apply H.
      Qed.

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen = witnessed_elements

      
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen = witnessed_elements

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen = witnessed_elements_old

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen βŠ† witnessed_elements_old
βŠ† eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen βŠ† witnessed_elements_old
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old βŠ† eval ρ patt_ind_gen

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
eval ρ patt_ind_gen βŠ† witnessed_elements_old
 apply interp_included_in_witnessed_elements_old.
        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
witnessed_elements_old βŠ† eval ρ patt_ind_gen
 apply witnessed_elements_old_included_in_interp.
      Qed.

      Section injective.
        Hypothesis (Domain_eq_dec : EqDecision (Domain M)).
        Hypothesis (Hstep_total_function : @is_total_function _ M step witnessed_elements witnessed_elements ρ).
        Hypothesis (Hstep_injective : @total_function_is_injective _ M step witnessed_elements ρ).

        Hypothesis (Hbase_step_no_confusion
                    : (eval ρ base)
                        ∩ (app_ext (eval ρ step) witnessed_elements) = βˆ…).

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚

        
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
is_witnessing_sequence m l₁
β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
(βˆ€ l₁ lβ‚‚ : list M,
   length lβ‚‚ ≀ length l₁
   β†’ is_witnessing_sequence m l₁
     β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚)
β†’ is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
length lβ‚‚ ≀ length l₁
β†’ is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
(βˆ€ l₁ lβ‚‚ : list M,
   length lβ‚‚ ≀ length l₁
   β†’ is_witnessing_sequence m l₁
     β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚)
β†’ is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
l₁ = lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
l: length l₁ ≀ length lβ‚‚
l₁ = lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
l₁ = lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
l: length l₁ ≀ length lβ‚‚
l₁ = lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
l: length l₁ ≀ length lβ‚‚
lβ‚‚ = l₁
 apply H; auto.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
l₁ = lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
length lβ‚‚ ≀ length l₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
is_witnessing_sequence m l₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
is_witnessing_sequence m lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
is_witnessing_sequence m l₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
is_witnessing_sequence m lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
H: βˆ€ l₁ lβ‚‚ : list M,
  length lβ‚‚ ≀ length l₁
  β†’ is_witnessing_sequence m l₁
    β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
n: Β¬ length l₁ ≀ length lβ‚‚
is_witnessing_sequence m lβ‚‚
 auto.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
length lβ‚‚ ≀ length l₁
β†’ is_witnessing_sequence m l₁
  β†’ is_witnessing_sequence m lβ‚‚ β†’ l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
l₁ = lβ‚‚


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
m ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
m ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
is_witnessing_sequence m l₁
 apply Hw₁. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
l₁ = lβ‚‚


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
lst₁: M
Hlst₁: last l₁ = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head l₁ = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip l₁ (tail l₁))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length []
lst₁: M
Hlst₁: last [] = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length [] lβ‚‚ = length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m, m₁: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head (m₁ :: l₁) = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length []
lst₁: M
Hlst₁: last [] = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length [] lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length []
lst₁: M
Hlst₁: None = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head [] = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length [] lβ‚‚ = length lβ‚‚
 inversion Hlst₁. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m, m₁: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: head (m₁ :: l₁) = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m, m₁: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: Some m₁ = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length (m₁ :: l₁) lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m, m₁: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m₁ :: l₁)
lst₁: M
Hlst₁: last (m₁ :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hhd₁: Some m₁ = Some m
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m₁ :: l₁) (tail (m₁ :: l₁)))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
H0: m₁ = m
common_length (m :: l₁) lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hhd₁: Some m = Some m
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) lβ‚‚ = length lβ‚‚


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last lβ‚‚ = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head lβ‚‚ = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip lβ‚‚ (tail lβ‚‚))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last [] = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head [] = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
mβ‚‚: M
lβ‚‚: list M
Hlen12: length (mβ‚‚ :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (mβ‚‚ :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head (mβ‚‚ :: lβ‚‚) = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (mβ‚‚ :: lβ‚‚) (tail (mβ‚‚ :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (mβ‚‚ :: lβ‚‚) = length (mβ‚‚ :: lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last [] = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head [] = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: None = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head [] = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) [] = length []
 inversion Hlstβ‚‚. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
mβ‚‚: M
lβ‚‚: list M
Hlen12: length (mβ‚‚ :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (mβ‚‚ :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: head (mβ‚‚ :: lβ‚‚) = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (mβ‚‚ :: lβ‚‚) (tail (mβ‚‚ :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (mβ‚‚ :: lβ‚‚) = length (mβ‚‚ :: lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
mβ‚‚: M
lβ‚‚: list M
Hlen12: length (mβ‚‚ :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (mβ‚‚ :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: Some mβ‚‚ = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (mβ‚‚ :: lβ‚‚) (tail (mβ‚‚ :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (mβ‚‚ :: lβ‚‚) = length (mβ‚‚ :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
mβ‚‚: M
lβ‚‚: list M
Hlen12: length (mβ‚‚ :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (mβ‚‚ :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hhdβ‚‚: Some mβ‚‚ = Some m
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (mβ‚‚ :: lβ‚‚) (tail (mβ‚‚ :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H0: mβ‚‚ = m
common_length (m :: l₁) (m :: lβ‚‚) = length (m :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hhdβ‚‚: Some m = Some m
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m :: lβ‚‚) = length (m :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (m :: l₁) (m :: lβ‚‚) = length (m :: lβ‚‚)


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
(if decide (m = m) then S (common_length l₁ lβ‚‚) else 0) =
S (length lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
e: m = m
S (common_length l₁ lβ‚‚) = S (length lβ‚‚)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
n: m β‰  m
0 = S (length lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
n: m β‰  m
0 = S (length lβ‚‚)
 contradiction. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
e: m = m
S (common_length l₁ lβ‚‚) = S (length lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
S (common_length l₁ lβ‚‚) = S (length lβ‚‚)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length (m :: lβ‚‚) ≀ length (m :: l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12': S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12': S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
length lβ‚‚ ≀ length l₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12': S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
Hlen12: length lβ‚‚ ≀ length l₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12': S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
length lβ‚‚ ≀ length l₁
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12': S (length lβ‚‚) ≀ S (length l₁)
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
Hlen12: length lβ‚‚ ≀ length l₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
Hlen12: length lβ‚‚ ≀ length l₁
common_length l₁ lβ‚‚ = length lβ‚‚


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ len₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ length l₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ length l₁
length l₁ ≀ len₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ len₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ length l₁
length l₁ ≀ len₁
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Heqlen₁: len₁ = length l₁
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ len₁
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
lst₁: M
Hlst₁: last (m :: l₁) = Some lst₁
Hbase₁: lst₁ ∈ eval ρ base
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
lstβ‚‚: M
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
len₁: nat
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ len₁
common_length l₁ lβ‚‚ = length lβ‚‚


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new ∈ app_ext (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ 0
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ 0
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] lβ‚‚ = length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (d :: l₁)
Hlen₁: length (d :: l₁) ≀ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) lβ‚‚ = length lβ‚‚

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (d :: l₁)
Hlen₁: length (d :: l₁) ≀ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length (d :: l₁)
Hlen₁: S (length l₁) ≀ 0
Hlst₁: last (m :: d :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: l₁) (tail (m :: d :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (d :: l₁) lβ‚‚ = length lβ‚‚
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] lβ‚‚ = length lβ‚‚

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
Hlen12: length [] ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last [m] = Some lstβ‚‚
Hfa₁, Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length [] [] = length []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] (d :: lβ‚‚) = length (d :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m: M
Hlen12: length [] ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last [m] = Some lstβ‚‚
Hfa₁, Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length [] [] = length []
 reflexivity.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] (d :: lβ‚‚) = length (d :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
length (d :: lβ‚‚) = 0
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H: length (d :: lβ‚‚) = 0
common_length [] (d :: lβ‚‚) = length (d :: lβ‚‚)

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
length (d :: lβ‚‚) = 0
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: S (length lβ‚‚) ≀ 0
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
length (d :: lβ‚‚) = 0
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H: length (d :: lβ‚‚) = 0
common_length [] (d :: lβ‚‚) = length (d :: lβ‚‚)

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
m, d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ 0
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: d :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: d :: lβ‚‚) (tail (m :: d :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H: length (d :: lβ‚‚) = 0
common_length [] (d :: lβ‚‚) = 0
 reflexivity.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ lβ‚‚ = length lβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last [m] = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length l₁ [] = length []
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
Hlen12: length (b :: lβ‚‚) ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last [m] = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
common_length l₁ [] = length []
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last [m] = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hmwit: m ∈ witnessed_elements
0 = length []
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
Hlen12: length (b :: lβ‚‚) ≀ length l₁
Hlen₁: length l₁ ≀ S len₁
Hlst₁: last (m :: l₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: l₁) (tail (m :: l₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length l₁ (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
Heq: l₁ = []
Hlen12: length (b :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ S len₁
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: lβ‚‚) = length (b :: lβ‚‚)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: a :: l'₁) (tail (m :: a :: l'₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
Heq: l₁ = []
Hlen12: length (b :: lβ‚‚) ≀ length []
Hlen₁: length [] ≀ S len₁
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
Heq: l₁ = []
Hlen12: S (length lβ‚‚) ≀ 0
Hlen₁: length [] ≀ S len₁
Hlst₁: last [m] = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip [m] (tail [m]))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length [] (b :: lβ‚‚) = length (b :: lβ‚‚)
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: a :: l'₁) (tail (m :: a :: l'₁)))
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)



              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m: M
l₁: list M
b: M
lβ‚‚: list M
a: M
l'₁: list M
Heq: l₁ = a :: l'₁
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
x: (M * M)%type
l: list (M * M)
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H: x = (m, a)
H0: l =
match l'₁ with
| [] => []
| x2 :: l2 => (a, x2) :: zip l'₁ l2
end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, a)
   :: match l'₁ with
      | [] => []
      | x2 :: l2 => (a, x2) :: zip l'₁ l2
      end)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (m :: b :: lβ‚‚) (tail (m :: b :: lβ‚‚)))
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match lβ‚‚ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match lβ‚‚ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
x: (M * M)%type
l: list (M * M)
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
H: x = (m, b)
H0: l =
match lβ‚‚ with
| [] => []
| x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((m, b)
   :: match lβ‚‚ with
      | [] => []
      | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
      end)
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
a ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
a ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
is_witnessing_sequence a (a :: l'₁)

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
βˆƒ lst : M,
  last (a :: l'₁) = Some lst ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
head (a :: l'₁) = Some a
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (a :: l'₁) (tail (a :: l'₁)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
βˆƒ lst : M,
  last (a :: l'₁) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
last (a :: l'₁) = Some lst₁ ∧ lst₁ ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
last (a :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
lst₁ ∈ eval ρ base

                  
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
last (a :: l'₁) = Some lst₁
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
last (a :: l'₁) = Some lst₁
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
last (a :: l'₁) = Some lst₁
 apply Hlst₁. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
lst₁ ∈ eval ρ base

                  apply Hbase₁.
                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
head (a :: l'₁) = Some a
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (a :: l'₁) (tail (a :: l'₁)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
head (a :: l'₁) = Some a
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l'₁) (tail (a :: l'₁)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
head (a :: l'₁) = Some a
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (a :: l'₁) (tail (a :: l'₁)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
 apply Hfa₁.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
b ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
is_witnessing_sequence b (b :: lβ‚‚)

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
βˆƒ lst : M,
  last (b :: lβ‚‚) = Some lst ∧ lst ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: lβ‚‚) = Some b
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
βˆƒ lst : M,
  last (b :: lβ‚‚) = Some lst ∧ lst ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: lβ‚‚) = Some lstβ‚‚ ∧ lstβ‚‚ ∈ eval ρ base
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
lstβ‚‚ ∈ eval ρ base

                  
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: lβ‚‚) = Some lstβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: lβ‚‚) = Some lstβ‚‚
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
last (b :: lβ‚‚) = Some lstβ‚‚
 apply Hlstβ‚‚. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
lstβ‚‚ ∈ eval ρ base

                  apply Hbaseβ‚‚.
                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: lβ‚‚) = Some b
∧ Forall
    (uncurry
       (Ξ» new old : M,
          new ∈ app_ext (eval ρ step) {[old]}))
    (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: lβ‚‚) = Some b
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
head (b :: lβ‚‚) = Some b
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
 apply Hfaβ‚‚.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, a)
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  m, b)
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
app_ext (eval ρ step) {[a]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
H: βˆƒ mβ‚‚ : M,
  mβ‚‚ ∈ witnessed_elements
  ∧ app_ext (eval ρ step) {[a]} = {[mβ‚‚]}
app_ext (eval ρ step) {[a]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
app_ext (eval ρ step) {[a]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
Hma: m ∈ {[a']}
app_ext (eval ρ step) {[a]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
a': M
Ha': app_ext (eval ρ step) {[a]} = {[a']}
Hma: m ∈ {[a']}
H: a' = m
app_ext (eval ρ step) {[a]} = {[m]}
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Hma: m ∈ {[m]}
Ha': app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[a]} = {[m]}

                apply Ha'.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
H: βˆƒ mβ‚‚ : M,
  mβ‚‚ ∈ witnessed_elements
  ∧ app_ext (eval ρ step) {[b]} = {[mβ‚‚]}
app_ext (eval ρ step) {[b]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
app_ext (eval ρ step) {[b]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
Hmb: m ∈ {[b']}
app_ext (eval ρ step) {[b]} = {[m]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
b': M
Hb': app_ext (eval ρ step) {[b]} = {[b']}
Hmb: m ∈ {[b']}
H: b' = m
app_ext (eval ρ step) {[b]} = {[m]}
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: βˆ€ m₁ : M,
  m₁ ∈ witnessed_elements
  β†’ βˆƒ mβ‚‚ : M,
      mβ‚‚ ∈ witnessed_elements
      ∧ app_ext (eval ρ step)
          {[m₁]} = {[mβ‚‚]}
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hmb: m ∈ {[m]}
Hb': app_ext (eval ρ step) {[b]} = {[m]}
app_ext (eval ρ step) {[b]} = {[m]}

                apply Hb'.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a = b
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
Haeqb: a = b
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a = b
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
a ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
b ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
b ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
rel_of ρ step a = rel_of ρ step b

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
app_ext (eval ρ step) {[a]} =
app_ext (eval ρ step) {[b]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
{[m]} = app_ext (eval ρ step) {[b]}

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
{[m]} = {[m]}

                reflexivity.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚: list M
a: M
l'₁: list M
Hlen12: length (b :: lβ‚‚) ≀ length (a :: l'₁)
Hlen₁: length (a :: l'₁) ≀ S len₁
Hlst₁: last (m :: a :: l'₁) = Some lst₁
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hma: m ∈ app_ext (eval ρ step) {[a]}
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (a, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita: a ∈ witnessed_elements
Hwitb: b ∈ witnessed_elements
Ham: app_ext (eval ρ step) {[a]} = {[m]}
Hbm: app_ext (eval ρ step) {[b]} = {[m]}
Haeqb: a = b
common_length (a :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hma, Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwita, Hwitb: b ∈ witnessed_elements
Ham, Hbm: app_ext (eval ρ step) {[b]} = {[m]}
common_length (b :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
common_length (b :: l'₁) (b :: lβ‚‚) = length (b :: lβ‚‚)


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
(if decide (b = b)
 then S (common_length l'₁ lβ‚‚)
 else 0) = S (length lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
S (common_length l'₁ lβ‚‚) = S (length lβ‚‚)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
n: b β‰  b
0 = S (length lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
n: b β‰  b
0 = S (length lβ‚‚)
 contradiction. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
S (common_length l'₁ lβ‚‚) = S (length lβ‚‚)

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
common_length l'₁ lβ‚‚ = length lβ‚‚


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length lβ‚‚ ≀ length l'₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≀ len₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length lβ‚‚ ≀ length l'₁
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: S (length lβ‚‚) ≀ S (length l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length lβ‚‚ ≀ length l'₁
  lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≀ len₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≀ len₁
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: S (length l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
length l'₁ ≀ len₁
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: l'₁) = Some lst₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
last (b :: lβ‚‚) = Some lstβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: l'₁) (tail (b :: l'₁)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements


              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  (zip (b :: lβ‚‚) (tail (b :: lβ‚‚)))
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
lst₁: M
Hbase₁: lst₁ ∈ eval ρ base
lstβ‚‚: M
Hbaseβ‚‚: lstβ‚‚ ∈ eval ρ base
len₁: nat
IHlen₁: βˆ€ (m : M) (l₁ lβ‚‚ : list M),
  length lβ‚‚ ≀ length l₁
  β†’ length l₁ ≀ len₁
    β†’ last (m :: l₁) = Some lst₁
      β†’ last (m :: lβ‚‚) = Some lstβ‚‚
        β†’ Forall
            (uncurry
               (Ξ» new old : M,
                  new
                  ∈ app_ext 
                      (eval ρ step) {[old]}))
            (zip (m :: l₁) (tail (m :: l₁)))
          β†’ Forall
              (uncurry
                 (Ξ» new old : M,
                    new
                    ∈ app_ext 
                        (eval ρ step) {[old]}))
              (zip (m :: lβ‚‚) (tail (m :: lβ‚‚)))
            β†’ m ∈ witnessed_elements
              β†’ common_length l₁ lβ‚‚ =
                length lβ‚‚
m, b: M
lβ‚‚, l'₁: list M
Hlst₁: last (m :: b :: l'₁) = Some lst₁
Hlen₁: length (b :: l'₁) ≀ S len₁
Hlen12: length (b :: lβ‚‚) ≀ length (b :: l'₁)
Hlstβ‚‚: last (m :: b :: lβ‚‚) = Some lstβ‚‚
Hmwit: m ∈ witnessed_elements
Hfa₁: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l'₁ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip l'₁ l2
  end
Hmb: m ∈ app_ext (eval ρ step) {[b]}
Hfaβ‚‚: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match lβ‚‚ with
  | [] => []
  | x2 :: l2 => (b, x2) :: zip lβ‚‚ l2
  end
Hwitb: b ∈ witnessed_elements
e: b = b
b ∈ witnessed_elements

              apply Hwitb.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
l₁ = lβ‚‚


          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
l₁ !! (length lβ‚‚ - 1) = lβ‚‚ !! (length lβ‚‚ - 1)
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
l₁ !! (length lβ‚‚ - 1) = lβ‚‚ !! (length lβ‚‚ - 1)

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
length lβ‚‚ - 1 < common_length l₁ lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
length lβ‚‚ - 1 < length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
length lβ‚‚ > 0
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
H: length lβ‚‚ > 0
length lβ‚‚ - 1 < length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
length lβ‚‚ > 0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m (d :: lβ‚‚)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: lβ‚‚) = length (d :: lβ‚‚)
length (d :: lβ‚‚) > 0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: (βˆƒ lst : M,
   last [] = Some lst ∧ lst ∈ eval ρ base)
∧ head [] = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip [] (tail []))
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: (βˆƒ lst : M, None = Some lst ∧ lst ∈ eval ρ base)
∧ None = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      []
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Contra: None = Some m
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ [] = length []
length [] > 0

                inversion Contra.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m (d :: lβ‚‚)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: lβ‚‚) = length (d :: lβ‚‚)
length (d :: lβ‚‚) > 0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
d: M
lβ‚‚: list M
Hlen12: length (d :: lβ‚‚) ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m (d :: lβ‚‚)
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ (d :: lβ‚‚) = length (d :: lβ‚‚)
S (length lβ‚‚) > 0
 lia.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
H: length lβ‚‚ > 0
length lβ‚‚ - 1 < length lβ‚‚

            lia.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
l₁ = lβ‚‚


          (*assert (Hbasem': eval ρ base lst).*)

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Β¬ length l₁ > length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Β¬ length l₁ > length lβ‚‚

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
length lβ‚‚ - 1 < length l₁
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
Hlt: length lβ‚‚ - 1 < length l₁
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
length lβ‚‚ - 1 < length l₁
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
Hlt: length lβ‚‚ - 1 < length l₁
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
Hlt: length lβ‚‚ - 1 < length l₁
Hexm: βˆƒ x : M, l₁ !! (length lβ‚‚ - 1) = Some x
False
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
Hexm: βˆƒ x : M, l₁ !! (length lβ‚‚ - 1) = Some x
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
Hwβ‚‚': is_witnessing_sequence m lβ‚‚
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
Hwβ‚‚': (βˆƒ lst : M,
   last lβ‚‚ = Some lst ∧ lst ∈ eval ρ base)
∧ head lβ‚‚ = Some m
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip lβ‚‚ (tail lβ‚‚))
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst1: last lβ‚‚ = Some lst
Hlst2: lst ∈ eval ρ base
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
m' = lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
Hlsteqm': m' = lst
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
m' = lst

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
Some m' = Some lst
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
H: Some m' = Some lst
m' = lst

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
Some m' = Some lst
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
l₁ !! (length lβ‚‚ - 1) = Some lst
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
l₁ !! (length lβ‚‚ - 1) = lβ‚‚ !! (length lβ‚‚ - 1)
 apply Hlast12.  
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
H: Some m' = Some lst
m' = lst

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
H: Some m' = Some lst
H1: m' = lst
lst = lst
 reflexivity.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
lst: M
Hlst2: lst ∈ eval ρ base
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some lst
Hlsteqm': m' = lst
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: m' ∈ eval ρ base
β†’ is_witnessing_sequence m'
    (drop (length lβ‚‚ - 1) l₁)
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some m'
Hlst2: m' ∈ eval ρ base
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hm': l₁ !! (length lβ‚‚ - 1) = Some m'
Hwsm: is_witnessing_sequence m'
  (drop (length lβ‚‚ - 1) l₁)
Hlst1: lβ‚‚ !! (length lβ‚‚ - 1) = Some m'
Hlst2: m' ∈ eval ρ base
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Hwsm: is_witnessing_sequence m'
  (drop (length lβ‚‚ - 1) l₁)
Hlst2: m' ∈ eval ρ base
False


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Heq: drop (length lβ‚‚ - 1) l₁ = []
Hwsm: is_witnessing_sequence m' []
Hlst2: m' ∈ eval ρ base
False
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: l
Hwsm: is_witnessing_sequence m' (d :: l)
Hlst2: m' ∈ eval ρ base
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Heq: drop (length lβ‚‚ - 1) l₁ = []
Hwsm: is_witnessing_sequence m' []
Hlst2: m' ∈ eval ρ base
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m': M
Heq: drop (length lβ‚‚ - 1) l₁ = []
Contra: head [] = Some m'
Hlst2: m' ∈ eval ρ base
False

              inversion Contra.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: l
Hwsm: is_witnessing_sequence m' (d :: l)
Hlst2: m' ∈ eval ρ base
False

            (* if `l` is empty, then length l₁ = length lβ‚‚ -> Contradiction *)
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
False
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length (drop (length lβ‚‚ - 1) l₁) = 1
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length (drop (length lβ‚‚ - 1) l₁) = 1
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length (drop (length lβ‚‚ - 1) l₁) = 1
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
length [d] = 1
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length (drop (length lβ‚‚ - 1) l₁) = 1
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
length lβ‚‚ β‰  0
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
H: length lβ‚‚ β‰  0
False

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
length lβ‚‚ β‰  0
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
Hcontra': length lβ‚‚ = 0
False

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
Hcontra': lβ‚‚ = []
False

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m []
Hmwit: m ∈ witnessed_elements
Hcontra: length l₁ > length []
Hlast12: l₁ !! (length [] - 1) =
[] !! (length [] - 1)
Hlcom: common_length l₁ [] = length []
m', d: M
Heq: drop (length [] - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length [] - 1) = 1
False

                
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁: list M
Hlen12: length [] ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
HContra'': head [] = Some m
Hmwit: m ∈ witnessed_elements
Hcontra: length l₁ > length []
Hlast12: l₁ !! (length [] - 1) =
[] !! (length [] - 1)
Hlcom: common_length l₁ [] = length []
m', d: M
Heq: drop (length [] - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length [] - 1) = 1
False

                inversion HContra''.
              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d: M
Heq: drop (length lβ‚‚ - 1) l₁ = [d]
Hwsm: is_witnessing_sequence m' [d]
Hlst2: m' ∈ eval ρ base
Hlendrop: length l₁ - (length lβ‚‚ - 1) = 1
H: length lβ‚‚ β‰  0
False

              lia.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
False

            (* `d0` is a member of `witnessed_elements` *)
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: head (tail (d :: d0 :: l)) = Some ?m'
β†’ is_witnessing_sequence 
    ?m' (tail (d :: d0 :: l))
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
β†’ is_witnessing_sequence ?m' (d0 :: l)
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
β†’ is_witnessing_sequence ?m' (d0 :: l)
Some d0 = Some d0
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
β†’ is_witnessing_sequence ?m' (d0 :: l)
Htmp: Some d0 = Some d0
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
β†’ is_witnessing_sequence ?m' (d0 :: l)
Some d0 = Some d0
 reflexivity. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: Some d0 = Some ?m'
β†’ is_witnessing_sequence ?m' (d0 :: l)
Htmp: Some d0 = Some d0
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Htmp: Some d0 = Some d0
False
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
False


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
d0 ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
d0 ∈ witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
is_witnessing_sequence d0 (d0 :: l)
 apply Hwd0. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False


            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
Heqm'd: m' = d
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: (βˆƒ lst : M,
   last (d :: d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (d :: d0 :: l) = Some m'
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (d :: d0 :: l) (tail (d :: d0 :: l)))
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
H: head (d :: d0 :: l) = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
H: Some d = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
m' = d
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
H: Some d = Some m'
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: d = m'
m' = m'

              reflexivity.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
m', d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hwsm: is_witnessing_sequence m' (d :: d0 :: l)
Hlst2: m' ∈ eval ρ base
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
Heqm'd: m' = d
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
False


            (* `m'=d` matches `app_ext (eval ρ step) d0` *)
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: (βˆƒ lst : M,
   last (d :: d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ head (d :: d0 :: l) = Some d
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      (zip (d :: d0 :: l) (tail (d :: d0 :: l)))
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: (βˆƒ lst : M,
   last (d0 :: l) = Some lst
   ∧ lst ∈ eval ρ base)
∧ Some d = Some d
  ∧ Forall
      (uncurry
         (Ξ» new old : M,
            new ∈ app_ext (eval ρ step) {[old]}))
      ((d, d0)
       :: match l with
          | [] => []
          | x2 :: l2 => (d0, x2) :: zip l l2
          end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
d ∈ app_ext (eval ρ step) {[d0]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
x: (M * M)%type
l0: list (M * M)
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  d, d0)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
H: x = (d, d0)
H0: l0 =
match l with
| [] => []
| x2 :: l2 => (d0, x2) :: zip l l2
end
d ∈ app_ext (eval ρ step) {[d0]}
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: uncurry
  (Ξ» new old : M,
     new ∈ app_ext (eval ρ step) {[old]}) (
  d, d0)
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[d0]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  ((d, d0)
   :: match l with
      | [] => []
      | x2 :: l2 => (d0, x2) :: zip l l2
      end)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H1: d ∈ app_ext (eval ρ step) {[d0]}
H2: Forall
  (uncurry
     (Ξ» new old : M,
        new ∈ app_ext (eval ρ step) {[old]}))
  match l with
  | [] => []
  | x2 :: l2 => (d0, x2) :: zip l l2
  end
d ∈ app_ext (eval ρ step) {[d0]}

              apply H1.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: eval ρ base
∩ app_ext (eval ρ step) witnessed_elements βŠ† βˆ…
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: βˆ€ x : M,
  x
  ∈ eval ρ base
    ∩ app_ext (eval ρ step) witnessed_elements
  β†’ x ∈ βˆ…
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
β†’ d ∈ βˆ…
False

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: False
False
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
β†’ d ∈ βˆ…
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: False
False
 exact H1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
H1: d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements
β†’ d ∈ βˆ…
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d
∈ eval ρ base
  ∩ app_ext (eval ρ step) witnessed_elements

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ eval ρ base
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements

            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ eval ρ base
 apply Hlst2.
            
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d ∈ app_ext (eval ρ step) witnessed_elements

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
H: d ∈ app_ext (eval ρ step) {[d0]}
d
∈ {[ e | βˆƒ le re : M,
           le ∈ eval ρ step
           ∧ re ∈ witnessed_elements
             ∧ e ∈ app_interp M le re ]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d
∈ {[ e | βˆƒ le re : M,
           le ∈ eval ρ step
           ∧ re ∈ witnessed_elements
             ∧ e ∈ app_interp M le re ]}

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
βˆƒ re : M,
  le' ∈ eval ρ step
  ∧ re ∈ witnessed_elements ∧ d ∈ app_interp M le' re
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
le' ∈ eval ρ step
∧ d0 ∈ witnessed_elements ∧ d ∈ app_interp M le' d0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
le' ∈ eval ρ step
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d0 ∈ witnessed_elements ∧ d ∈ app_interp M le' d0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
le' ∈ eval ρ step
 apply H1. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d0 ∈ witnessed_elements ∧ d ∈ app_interp M le' d0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d0 ∈ witnessed_elements
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d ∈ app_interp M le' d0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d0 ∈ witnessed_elements
 apply Hd0we. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
d ∈ app_interp M le' d0

              
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d, d0: M
l: list M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: d0 :: l
Hlst2: d ∈ eval ρ base
Hwsm: is_witnessing_sequence d (d :: d0 :: l)
Hwd0: is_witnessing_sequence d0 (d0 :: l)
Hd0we: d0 ∈ witnessed_elements
le', re': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[d0]}
H3: d ∈ app_interp M le' re'
H: d0 = re'
d ∈ app_interp M le' re'
 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements βŠ† βˆ…
βŠ† eval ρ base
  ∩ app_ext (eval ρ step)
      witnessed_elements
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hcontra: length l₁ > length lβ‚‚
d: M
l: list M
re': M
Heq: drop (length lβ‚‚ - 1) l₁ = d :: re' :: l
Hlst2: d ∈ eval ρ base
Hd0we: re' ∈ witnessed_elements
Hwd0: is_witnessing_sequence re' (re' :: l)
Hwsm: is_witnessing_sequence d (d :: re' :: l)
le': M
H1: le' ∈ eval ρ step
H2: re' ∈ {[re']}
H3: d ∈ app_interp M le' re'
d ∈ app_interp M le' re'
 apply H3.
          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
length l₁ = length lβ‚‚
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
Hlength12: length l₁ = length lβ‚‚
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
length l₁ = length lβ‚‚
 lia. 
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hlen12: length lβ‚‚ ≀ length l₁
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
H: Β¬ length l₁ > length lβ‚‚
Hlength12: length l₁ = length lβ‚‚
l₁ = lβ‚‚

          
Ξ£: Signature
base, step: Pattern
patt_ind_gen_body:= nest_mu base
or nest_mu step $ BoundVarSugar.B0: Pattern
patt_ind_gen_simple_body:= base
or step $
   patt_free_svar
     (fresh_svar
        patt_ind_gen_body): Pattern
Hwfpbase: well_formed_positive base
Hwfpstep: well_formed_positive step
M: Model
ρ: Valuation
F:= let X := fresh_svar patt_ind_gen_body in
Fassoc ρ patt_ind_gen_body^{svar:0↦X} X: propset M β†’ propset M
Domain_eq_dec: EqDecision M
Hstep_total_function: is_total_function step
  witnessed_elements
  witnessed_elements ρ
Hstep_injective: total_function_is_injective step
  witnessed_elements ρ
Hbase_step_no_confusion: eval ρ base
∩ app_ext (eval ρ step)
    witnessed_elements = βˆ…
m: M
l₁, lβ‚‚: list M
Hw₁: is_witnessing_sequence m l₁
Hwβ‚‚: is_witnessing_sequence m lβ‚‚
Hmwit: m ∈ witnessed_elements
Hlcom: common_length l₁ lβ‚‚ = length lβ‚‚
Hlast12: l₁ !! (length lβ‚‚ - 1) =
lβ‚‚ !! (length lβ‚‚ - 1)
Hlength12: length l₁ = length lβ‚‚
l₁ = lβ‚‚


          apply (common_length_impl_eq _ _ Hlength12 Hlcom).
        Qed.

      End injective.

    End with_eval.

  End inductive_generation.

End with_signature.